Adaptive Metric Convergence: A New Framework Beyond Weak Topologies

1. Introduction

The notion of distance sits at the very heart of modern analysis. Whether we work with the rigid structures of Banach spaces or the more flexible geometries of Riemannian manifolds, the choice of a metric shapes everything: how we define convergence, what compactness means, and what continuity looks like. Yet, in nearly every classical setting, the metric is chosen in advance and never changes during the analysis. The distance between two points depends only on the points themselves and on a fixed ruler. This idea, the fixity of the metric, is so deeply embedded in the way we think that we rarely stop to question it.

This paper questions it. We develop a framework where the distance between two objects is not fixed. Instead, it depends on the first object itself, through a measure that adapts to that object. This creates a feedback loop: the metric we use to measure closeness is shaped by the reference point, and that metric in turn affects how the system evolves. We call this phenomenon discordant convergence. The name reflects the fact that the notion of closeness becomes self-referential and, in a precise sense, non-topological.

The idea of metrics that vary is not completely new. In differential geometry, Finsler manifolds allow the norm to depend on direction, but the base point is only a parameter; the distance function itself remains fixed [3,15]. In functional analysis, quasi-metric spaces allow asymmetric distances, but the asymmetry is fixed once and for all [7,11]. In machine learning, metric learning adjusts distances from data, but once the training is done, the metric stays fixed for inference [4,17]. In numerical analysis, adaptive step-size methods change the resolution locally, but the underlying norm does not change [10]. Even in homogenization theory—which transformed the study of multiscale problems by breaking the separation of scales—the metrics at the microscopic and macroscopic levels are fixed from the start [5,16].

Our approach is different. We introduce a family of measures ${\left({\mu }_f\right)}_{f\in X}$, indexed by the elements of the space itself. To measure the distance between g and h, we use the weight ${\mu }_f$, where f is the reference point typically the first argument. For each fixed f, the distance $D_f(g,h)$ is a classical metric. But when we look at convergence of a sequence $\left(u_n\right)$, the rule is $D_{u_n}(u_n,u)\to 0$. The metric moves with the sequence. This change may seem small, but it has far-reaching consequences. The convergence structure that emerges is not induced by any topology. Closed balls are not sequentially closed. The usual tools based on open sets no longer apply.

This paper builds the rigorous foundations for this new way of thinking. The main contributions are the following:

1.

We define abstract structures: discordant normed spaces, discordant Banach spaces, and discordant Hilbert spaces. These extend classical functional analysis in a natural way.

2.

We give concrete examples using weighted $L^1$ and $L^2$ Sobolev spaces. The weight is determined by the derivative of the reference function. We prove that these examples are complete, and that they have natural discordant versions of reflexivity and separability.

3.

We prove a compactness theorem for bounded sequences in $W^{1,1}(0,1)$: every bounded sequence has a discordantly convergent subsequence. This is the discordant analogue of the classical Bolzano-Weierstrass theorem.

4.

We show that the framework is useful by solving a nonlinear parabolic equation where the reaction term depends on the discordant distance. The proof uses a fixed-point argument and relies on our compactness theorem.

The paper is organised as follows. Section 2 collects the basic tools: function spaces, mollifiers, and the Dunford-Pettis theorem. Section 3 introduces the discordant distance and convergence, and proves that the associated balls do not form a topology base. Section 4 sets up the abstract discordant Banach and Hilbert spaces. Section 5 gives a full proof of completeness for the weighted $L^2$ example. Section 6 defines and proves discordant reflexivity and separability. Section 7 applies the theory to a parabolic equation with adaptive feedback.

To the best of our knowledge, this is the first systematic treatment of convergence structures where the metric adapts to the sequence itself. By freeing the distance from a fixed reference, we open the way to a new area of analysis where the observer and the observed are linked, and where the notion of proximity is itself a dynamic quantity.

2. Preliminaries

The purpose of this section is to lay out the classical tools that will support our discordant structures. We gather together the function spaces, approximation results, compactness criteria, and the adaptive measures that will form the backbone of what follows. Readers who are already familiar with functional analysis can go straight to the definitions of the adaptive measures. Still, we do recommend reading through the whole section. The reason is simple: the novelty of our work lies precisely in the way classical properties interact with their discordant counterparts. Understanding that interplay is key to appreciating what comes later.

2.1. Function Spaces and Structural Embeddings

Let $I=[0,1]$ be the unit interval, equipped with the Lebesgue measure $dx$. For $1\le p\le \infty$, the Lebesgue space $L^p\left(I\right)$ consists of equivalence classes of measurable functions f for which

$$\begin{array}{c}\hfill {\displaystyle {\parallel f\parallel }_{L^p}\equiv \left\{\begin{array}{cc}{\displaystyle {\left({\int }_I{\left|f\left(x\right)\right|}^{p}dx\right)}^{1/p},}\hfill & 1\le p<\infty ,\hfill \\ {\displaystyle {ess\; sup}_{x\in I}\left|f\left(x\right)\right|,}\hfill & p=\infty ,\hfill \end{array}\right.}\end{array}$$

is finite. These spaces are complete (Banach spaces), and for $1<p<\infty$, they are reflexive. The case $p=1$ is non-reflexive, which will require special care in our compactness arguments.

We shall frequently work with two Sobolev spaces. The first is

$$\begin{array}{c}\hfill {\displaystyle W^{1,1}\left(I\right)\equiv \left\{f\in L^1\left(I\right)|\mathrm{there}\mathrm{exists}{f}^{\prime }\in L^1\left(I\right)\mathrm{such}\mathrm{that}{\int }_{I}f{\phi }^{\prime }dx=-{\int }_I{f}^{\prime }\phi dx\forall \phi \in C_c^{\infty }\left(I\right)\right\}.}\end{array}$$

In other words, $W^{1,1}\left(I\right)$ is the space of integrable functions whose distributional derivative is again integrable. Endowed with the norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_{W^{1,1}}\equiv {\parallel f\parallel }_{L^1}+{\parallel f^{\prime }\parallel }_{L^1},\end{array}$$

it becomes a Banach space. A fundamental property in one dimension is that every $f\in W^{1,1}\left(I\right)$ admits an absolutely continuous representative on I, which implies the embedding

$$\begin{array}{c}\hfill W^{1,1}\left(I\right)\hookrightarrow L^{\infty }\left(I\right)\end{array}$$

is continuous. Indeed, by the fundamental theorem of calculus [6,9], for any $x,y\in I$,

$$\begin{array}{c}\hfill \left|f\left(x\right)-f\left(y\right)\right|\le {\int }_I|f^{\prime }\left(t\right)|dt=\parallel f^{\prime }{\parallel }_{L^1},\end{array}$$

so ${\parallel f\parallel }_{L^{\infty }}\le {\parallel f\parallel }_{L^1}+{\parallel f^{\prime }\parallel }_{L^1}$. This seemingly elementary fact will be indispensable: it ensures that sequences bounded in $W^{1,1}$ are uniformly bounded in the $L^{\infty }$ norm, a property that cannot be taken for granted in higher dimensions.

The second space is the classical Hilbertian Sobolev space

$$\begin{array}{c}\hfill H^1\left(I\right)\equiv \left\{f\in L^2\left(I\right)|f^{\prime }\in L^2\left(I\right)\right\},\end{array}$$

with the inner product ${\langle f,g\rangle }_{H^1}\equiv {\langle f,g\rangle }_{L^2}+{\langle f^{\prime },g^{\prime }\rangle }_{L^2}$ and the induced norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_{H^1}\equiv {\left({\parallel f\parallel }_{L^2}^2+{\parallel f^{\prime }\parallel }_{L^2}^2\right)}^{1/2}.\end{array}$$

As with $W^{1,1}$, functions in $H^1\left(I\right)$ are absolutely continuous, so the embedding $H^1\left(I\right)\hookrightarrow L^{\infty }\left(I\right)$ is also continuous. This regularity, exclusive to the one-dimensional setting will allow us to control pointwise values of our sequences, a luxury we exploit heavily in the compactness proof.

A crucial distinction between these spaces lies in their compactness properties. The Rellich-Kondrachov theorem states that the embeddings

$$\begin{array}{c}\hfill W^{1,1}\left(I\right)\hookrightarrow L^1\left(I\right),H^1\left(I\right)\hookrightarrow L^2\left(I\right)\end{array}$$

are compact. In plain terms, any bounded sequence in the larger Sobolev space admits a subsequence that converges strongly in the corresponding Lebesgue space. This is the classical route to extracting limits from oscillatory sequences, and it will form the backbone of our discordant compactness theorem.

2.2. Mollification and Approximation

To bridge the gap between rough functions and smooth ones, we employ the standard technique of mollification. Let $\rho \in C_c^{\infty }\left(\mathbb{R}\right)$ be a non-negative, radially symmetric function supported in $[-1,1]$ with ${\displaystyle {\int }_{\mathbb{R}}\rho \left(x\right)dx=1}$. For $\epsilon >0$, define the scaled mollifier

$$\begin{array}{c}\hfill {\rho }_{\epsilon }\left(x\right)\equiv {\displaystyle \frac{1}{\epsilon }}\rho \left({\displaystyle \frac{x}{\epsilon }}\right).\end{array}$$

For a function $f\in L^1\left(I\right)$, we extend it by zero outside I and define the convolution

$$\begin{array}{c}\hfill f_{\epsilon }\left(x\right)\equiv (f\ast {\rho }_{\epsilon })\left(x\right)={\int }_{\mathbb{R}}f\left(y\right){\rho }_{\epsilon }(x-y)dy.\end{array}$$

The following classical approximation properties will be used repeatedly:

• $f_{\epsilon }\in C^{\infty }\left(\overline{I}\right)$ for every $\epsilon >0$;
• $f_{\epsilon }\to f$ in $L^p\left(I\right)$ for every $1\le p<\infty$ as $\epsilon \to 0^{+}$;
• If $f\in W^{1,1}\left(I\right)$ (resp. $H^1\left(I\right)$), then ${\left(f_{\epsilon }\right)}^{\prime }=f^{\prime }\ast {\rho }_{\epsilon }$, and hence $f_{\epsilon }\to f$ in the Sobolev norm.

Mollification provides a convenient way to approximate arbitrary Sobolev functions by smooth ones, which is essential when defining measures that depend on derivatives, as we shall do below. Moreover, the uniform boundedness of the $L^1$ norm of the derivatives of the mollified sequence $\left(f_{\epsilon }\right)$ (by the $L^1$ norm of $f^{\prime }$) will be crucial in extracting weak-* convergent subsequences of measures.

2.3. Weak Compactness in $L^1$: The Dunford-Pettis Theorem

Since $L^1\left(I\right)$ is not reflexive, bounded sequences do not have weakly convergent subsequences. The classical counterexample is $f_n\left(x\right)=n{\mathbf{1}}_{[0,1/n]}\left(x\right)$, which is bounded in $L^1$ but has no weakly convergent subsequence. The Dunford-Pettis theorem provides the precise characterization of weak compactness.

A family $\mathcal{F}\subset L^1\left(I\right)$ is said to be uniformly integrable if for every $\epsilon >0$ there exists $\delta >0$ such that for every measurable set $A\subset I$ with $\left|A\right|<\delta$ and every $f\in \mathcal{F}$,

$$\begin{array}{c}\hfill {\int }_A\left|f\left(x\right)\right|dx<\epsilon .\end{array}$$

Equivalently, uniform integrability prevents the mass of the functions from concentrating on sets of arbitrarily small measure.

Theorem 2.1.

([6,9], Dunford-Pettis) A bounded sequence in $L^1\left(I\right)$ admits a weakly convergent subsequence if and only if it is uniformly integrable.

In our discordant framework, we will frequently encounter sequences of derivatives $\left(u_n^{\prime }\right)$ that are bounded in $L^1$. By passing to a subsequence, we may assume they are uniformly integrable, ensuring weak convergence in $L^1$. This weak convergence, combined with the compact embeddings mentioned above, will allow us to pass to limits in the non-linear terms involving the adaptive measures.

2.4. Measures and Weak-* Convergence

Let $\mathcal{M}\left(I\right)$ denote the space of finite signed Borel measures on I. By the Riesz-Markov-Kakutani representation theorem [6,9], $\mathcal{M}\left(I\right)$ is isometrically isomorphic to the dual of $C\left(I\right)$, the space of continuous functions on I equipped with the supremum norm. A sequence $\left({\mu }_n\right)\subset \mathcal{M}\left(I\right)$ converges weak-* to $\mu$ if for every continuous test function $\phi \in C\left(I\right)$,

$$\begin{array}{c}\hfill {\int }_I\phi d{\mu }_n⟶{\int }_I\phi d\mu .\end{array}$$

The Banach-Alaoglu theorem (see e.g. [6]) ensures that the unit ball of $\mathcal{M}\left(I\right)$ is weak-* compact. Since the total variation of our adaptive measures will always be one (they are probability measures), we will be able to extract weak-* convergent subsequences straightforwardly. This compactness in the space of measures will be essential when studying the limit of the weights ${\mu }_{u_n}$ as $n\to \infty$.

2.5. The Adaptive Measures

We now introduce the two families of probability measures that encode the adaptive nature of our distances. The choice of weighting is not arbitrary; it is designed to make the measure sensitive to the local variation of the reference function.

For a function $f\in W^{1,1}\left(I\right)$, we define

$$d{\mu }_f\left(x\right)\equiv {\displaystyle \frac{1+|f^{\prime }\left(x\right)|}{1+\parallel f^{\prime }{\parallel }_{L^1}}}dx.$$

(2.1)

This is indeed a probability measure on I, since

$$\begin{array}{c}\hfill {\displaystyle {\int }_{I}d{\mu }_f=\frac{{\int }_{I}1dx+{\int }_I\left|f^{\prime }\right|dx}{1+\parallel f^{\prime }{\parallel }_{L^1}}=\frac{1+\parallel f^{\prime }{\parallel }_{L^1}}{1+\parallel f^{\prime }{\parallel }_{L^1}}=1.}\end{array}$$

Observe that if f has a large derivative in some region, that region receives more weight. Thus, when measuring the distance between two functions using ${\mu }_f$, we are effectively penalizing discrepancies in regions where f oscillates rapidly. This creates a natural feedback: the regularity of the reference point dictates the regions of interest for the comparison.

For the Hilbertian setting, we require a weighted $L^2$-norm. For $f\in H^1\left(I\right)$, define

$$d{\mu }_f\left(x\right)\equiv {\displaystyle \frac{1+|f^{\prime }\left(x\right){|}^2}{1+\parallel f^{\prime }{\parallel }_{L^2}^2}}dx.$$

(2.2)

The normalization is again exact:

$$\begin{array}{c}\hfill {\int }_{I}d{\mu }_f={\displaystyle \frac{1+\parallel f^{\prime }{\parallel }_{L^2}^2}{1+\parallel f^{\prime }{\parallel }_{L^2}^2}}=1.\end{array}$$

The use of the squared derivative here is natural for Hilbert-space geometry, as it aligns with the $L^2$-based Sobolev structure and will allow us to define an inner product that depends on the reference point.

For these two families of measures, one tailored to the $L^1$-Banach setting, the other to the $L^2$-Hilbert setting. They are the engines of our discordant theory. They are not merely technical devices; they embody the philosophy that the metric should adapt to the object being measured. The denominators are nothing more than normalization constants, but they ensure that each ${\mu }_f$ is a genuine probability measure, which is crucial for the compactness arguments involving weak-* limits.

With these tools in hand, we are now prepared to define the discordant distance and explore its radical departure from classical analysis.

3. Discordant Distance and Convergence

We are now ready to introduce the two central objects of our theory. The first is a family of distances, each one tied to a reference point. The second is a new notion of convergence that comes from the self-referential way these distances work. In this section, we set down the basic definitions and show that they lead to something genuinely new. The convergence we define is not induced by any topology. It is a purely sequential phenomenon.

3.1. The Discordant Distance

For each fixed reference function f, the measure ${\mu }_f$ assigns greater weight to regions where f varies rapidly. This allows us to define a distance that emphasizes discrepancies in those regions.

Definition 3.1

(Discordant distance). Let X denote either the Banach space $W^{1,1}\left(I\right)$ or the Hilbert space $H^1\left(I\right)$. For $f,g,h\in X$, define the discordant distance between g and h with respect to the reference f by

$$\begin{array}{c}\hfill {\displaystyle D_f(g,h)\equiv {\int }_I\left|g\left(x\right)-h\left(x\right)\right|d{\mu }_f\left(x\right),}\end{array}$$

where ${\mu }_f$ is given by (2.1) in the $W^{1,1}$ case, and by (2.2) in the $H^1$ case.

We pause to examine the structure of this object. For a fixed reference function f, the map

$$\begin{array}{c}\hfill (g,h)⟼D_f(g,h)\end{array}$$

satisfies all the axioms of a metric on X: it is symmetric, non-negative, satisfies the triangle inequality, and $D_f(g,h)=0$ implies $g=h$ almost everywhere (since the density of ${\mu }_f$ is strictly positive on I, up to a null set). Thus, for each fixed f, $(X,D_f)$ is a metric space.

However, the notation $D_f$ deliberately emphasizes the role of f as the reference. The novelty does not lie in the fact that $D_f$ is a metric for fixed f that is standard. The novelty lies in what happens when f is allowed to vary, particularly when it is chosen to be the first argument of the distance itself. This self-referentiality is the engine of discordant convergence.

Remark 3.1

(Asymmetry). An immediate and profound consequence of the definition is that the discordant distance is generally asymmetric:

$$\begin{array}{c}\hfill D_f(g,h)\ne D_g(g,h)\mathrm{in}\mathrm{general}.\end{array}$$

Indeed, the left-hand side weighs the discrepancy $\left|g-h\right|$ according to ${\mu }_f$, while the right-hand side uses ${\mu }_g$. Since ${\mu }_f$ and ${\mu }_g$ may differ dramatically for instance, if f has a large derivative where g does not, the two distances do not coincide. This asymmetry is not an artefact; it is a deliberate feature that encodes the idea that the measure of closeness depends on the observer.

3.2. Discordant Convergence

We now define the central convergence notion of this paper. Unlike classical convergence, where the metric is fixed once and for all, discordant convergence allows the metric to evolve with the sequence.

Definition 3.2

(Discordant convergence). A sequence ${\left(u_n\right)}_{n\in \mathbb{N}}\subset X$ is said to converge discordantly to a limit $u\in X$ if

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{D}_{u_n}(u_n,u)=0.\end{array}$$

In this case, we write

$$\begin{array}{c}\hfill u_n\stackrel{D}{\to }u.\end{array}$$

Let us parse this definition carefully. For each n, the distance between $u_n$ and u is measured using the measure ${\mu }_{u_n}$, which is determined by $u_n$ itself. Thus, as the sequence evolves, the ruler with which we measure proximity changes. This is a fundamental departure from classical analysis: there is no single metric that governs convergence. Instead, the sequence generates its own sequence of metrics.

To appreciate the depth of this departure, consider the classical notion of weak convergence in $L^p$: $u_n\rightharpoonup u$ if ${\displaystyle \int u_n\phi \to \int u\phi }$ for every test function $\phi$. This is a linear notion, independent of the sequence. In contrast, discordant convergence is inherently non-linear and adaptive: the test of closeness is weighted by the sequence itself.

Remark 3.2

(Comparison with classical convergence). Suppose, for a moment, that the measures ${\mu }_f$ were independent of f, ${\mu }_f=dx$ for all f. Then $D_f(g,h)={\parallel g-h\parallel }_{L^1}$, and discordant convergence would reduce to classical $L^1$ convergence. The entire novelty of our framework lies precisely in the dependence of ${\mu }_f$ on f. This dependence introduces a feedback loop that cannot be captured by any fixed topology.

3.3. Topological Nature: The Failure of Open Balls

A natural question arises: is discordant convergence induced by a topology? In classical analysis, convergence is typically topological: there exists a collection of open sets such that $u_n\to u$ if and only if every open neighbourhood of u contains all but finitely many $u_n$. For discordant convergence, one might hope to define a topology using the family of balls

$$\begin{array}{c}\hfill {\displaystyle \mathcal{B}(u,r)\equiv \left\{v\in X|D_u(u,v)<r\right\},}\end{array}$$

where u is the centre and $r>0$ is the radius. If these balls could serve as a base for a topology, then discordant convergence would be topological after all.

We now show that this is impossible. The family $\left\{\mathcal{B}\right(u,r):u\in X,r>0\}$ fails to satisfy the most basic requirement of a topology base: the intersection property.

Proposition 3.1.

Let X be either $W^{1,1}\left(I\right)$ or $H^1\left(I\right)$ with the discordant distance defined above. The family of balls $\left\{\mathcal{B}\right(u,r):u\in X,r>0\}$ is  not  a base for any topology on X.

Proof. 

We construct two balls whose intersection contains the zero function, yet no ball centred at zero is contained in their intersection. This violates the standard base axiom: for any point in the intersection of two base elements, there must be a base element containing that point and contained in the intersection.

For clarity, we work in the $W^{1,1}$ setting with the $L^1$-weighted distance; the argument is identical in the $H^1$ case with $|f^{\prime }{|}^2$ replacing $|f^{\prime }|$, and we comment on the necessary adjustments at the end.

Let $u_1\equiv 0$ be the zero function. Then $u_1^{\prime }\equiv 0$, so by (2.1), ${\mu }_{u_1}=dx$ and consequently

$$\begin{array}{c}\hfill D_{u_1}(u_1,v)={\int }_I\left|v\left(x\right)\right|dx={\parallel v\parallel }_{L^1}.\end{array}$$

Thus, the ball $\mathcal{B}(0,1)$ is precisely the open unit ball of $L^1\left(I\right)$.

Now fix a parameter $h>0$ to be chosen shortly. Let $E\subset I$ be an interval of small length $\alpha >0$. We shall construct a function $f\in W^{1,1}\left(I\right)$ supported in E such that:

1.

$0\le f\left(x\right)\le h$ for all $x\in E$ (so f has height at most h);

2.

${\displaystyle {\int }_{E}f\left(x\right)dx}$ is small (in fact, $\le h\alpha$);

3.

the $L^1$-norm of $f^{\prime }$ is arbitrarily large, say $\parallel f^{\prime }{\parallel }_{L^1}=M$, where M is a large constant to be chosen.

Such a function can be constructed by taking a slowly varying positive base of height $h/2$ (to ensure the integral is not too large) and adding a highly oscillatory component with zero mean and large derivative. For instance, on E, set

$$\begin{array}{c}\hfill f\left(x\right)={\displaystyle \frac{h}{2}}+\epsilon sin\left({\displaystyle \frac{2\pi Nx}{\alpha }}\right),\end{array}$$

where N is chosen so that $\parallel f^{\prime }{\parallel }_{L^1}\approx 2\pi \epsilon N$ is large, and $\epsilon$ is small enough so that $f\left(x\right)\le h$ for all x (e.g., $\epsilon \le h/4$). Then ${\int }_{E}f\left(x\right)dx\approx {\displaystyle \frac{h\alpha }{2}}$, and $\parallel f^{\prime }{\parallel }_{L^1}\approx 2\pi \epsilon N=:M$, which can be made arbitrarily large by increasing N. We choose M such that

$$\begin{array}{c}\hfill {\displaystyle \frac{M}{1+M}}>{\displaystyle \frac{2}{h}}\mathrm{and}{\displaystyle \frac{h}{2}}·{\displaystyle \frac{M}{1+M}}<1.\end{array}$$

The first inequality ensures that $(4-h){\mu }_f\left(E\right)>1$; the second ensures that $D_f(f,0)<1$. We can satisfy both by taking h small, say $h=0.5$, and M sufficiently large. For $h=0.5$, the first inequality becomes ${\displaystyle \frac{M}{1+M}}>4$, which is impossible since ${\displaystyle \frac{M}{1+M}}<1$. So we need to adjust the constant 4: we are free to choose the constant value of v on E. Let us denote by c the constant value of v on E (instead of 4). We want

$$\begin{array}{c}\hfill (c-h){\mu }_f\left(E\right)>1\mathrm{and}h{\mu }_f\left(E\right)<1,\end{array}$$

while also ${\parallel v\parallel }_{L^1}=c\alpha <t$ for arbitrary $t>0$. Since t is arbitrary, we can choose c as large as we want and $\alpha$ so small that $c\alpha <t$. Thus, we are free to take c arbitrarily large. For instance, choose $c=10$ and $h=0.5$. Then we need

$$\begin{array}{c}\hfill (10-0.5){\mu }_f\left(E\right)>1⟺9.5{\mu }_f\left(E\right)>1⟺{\mu }_f\left(E\right)>{\displaystyle \frac{1}{9.5}}\approx 0.1053.\end{array}$$

And we also need $0.5{\mu }_f\left(E\right)<1$, which is automatically true since ${\mu }_f\left(E\right)\le 1$. So we only need ${\mu }_f\left(E\right)>0.1053$. But ${\mu }_f\left(E\right)={\displaystyle \frac{\alpha +\parallel f^{\prime }{\parallel }_{L^1}}{1+\parallel f^{\prime }{\parallel }_{L^1}}}\approx {\displaystyle \frac{M}{1+M}}$, which tends to 1 as $M\to \infty$. Hence we can choose M large enough so that ${\mu }_f\left(E\right)>0.2$. Then the inequalities are satisfied.

To avoid technicalities with the oscillatory function, we give an explicit piecewise linear construction that yields the same estimates. Let $E=[a,a+\alpha ]$ with $\alpha$ very small. Divide E into $2N$ equal subintervals of length $\alpha /\left(2N\right)$. On each subinterval, define f to have a "sawtooth" pattern: it oscillates between 0 and h with linear slopes of magnitude $2Nh/\alpha$. The total variation of f over E is then $2Nh$ (each oscillation contributes $2h$), so $\parallel f^{\prime }{\parallel }_{L^1}=2Nh$. Meanwhile, the integral of f over E is exactly $h\alpha /2$ (since the average value over each oscillation is $h/2$). Thus, ${\displaystyle {\int }_{E}f=h\alpha /2}$, which is small. We choose N large enough so that $M\equiv 2Nh$ satisfies

$$\begin{array}{c}\hfill {\displaystyle \frac{M}{1+M}}>{\displaystyle \frac{2}{c-h}}\end{array}$$

where c is the constant value of v on E to be chosen later.

Now choose $h=0.5$ and $c=10$. Then $c-h=9.5$, and we need ${\displaystyle \frac{M}{1+M}}>{\displaystyle \frac{2}{9.5}}\approx 0.2105$. Taking $M=10$ gives ${\displaystyle \frac{10}{11}}\approx 0.909>0.2105$, which is more than enough. With this choice, we have

$$\begin{array}{c}\hfill {\mu }_f\left(E\right)={\displaystyle \frac{{\int }_E(1+|f^{\prime }\left|\right)dx}{1+\parallel f^{\prime }{\parallel }_{L^1}}}={\displaystyle \frac{\alpha +M}{1+M}}.\end{array}$$

For $\alpha$ sufficiently small (e.g., $\alpha <1$), we have ${\mu }_f\left(E\right)\ge {\displaystyle \frac{M}{1+M}}>0.2$.

Now compute the discordant distance from f to 0:

$$\begin{array}{c}\hfill D_f(f,0)={\int }_I\left|f\left(x\right)\right|d{\mu }_f\left(x\right)={\int }_{E}f\left(x\right)d{\mu }_f\left(x\right)\le h{\mu }_f\left(E\right)=0.5\times 0.2=0.1<1,\end{array}$$

so $0\in \mathcal{B}(f,1)$. Trivially $0\in \mathcal{B}(0,1)$, hence

$$\begin{array}{c}\hfill 0\in \mathcal{B}(0,1)\cap \mathcal{B}(f,1).\end{array}$$

Now let $t>0$ be arbitrary. Choose $\alpha <t/c=t/10$. Define v to be constant equal to $c=10$ on E and 0 elsewhere:

$$\begin{array}{c}\hfill v\left(x\right)\equiv 10·{\mathbf{1}}_E\left(x\right).\end{array}$$

Then ${\parallel v\parallel }_{L^1}=10\alpha <t$, so $v\in \mathcal{B}(0,t)$.

However, on E, we have $f\left(x\right)\le h=0.5$, so $\left|f\left(x\right)-v\left(x\right)\right|=v\left(x\right)-f\left(x\right)\ge 10-0.5=9.5$ for all $x\in E$. Therefore

$$\begin{array}{c}\hfill D_f(f,v)={\int }_I\left|f\left(x\right)-v\left(x\right)\right|d{\mu }_f\left(x\right)\ge {\int }_{E}9.5d{\mu }_f\left(x\right)=9.5{\mu }_f\left(E\right)\ge 9.5\times 0.2=1.9>1.\end{array}$$

Thus, $v\notin \mathcal{B}(f,1)$.

Since $t>0$ was arbitrary, no ball $\mathcal{B}(0,t)$ can be contained in the intersection $\mathcal{B}(0,1)\cap \mathcal{B}(f,1)$: for every $t>0$, the function v constructed above lies in $\mathcal{B}(0,t)$ but outside the intersection. Hence, the family of balls fails the base axiom for a topology.

The same construction works in the $H^1$ case with the $L^2$-weighted measure, by replacing $|f^{\prime }|$ with $|f^{\prime }{|}^2$ and choosing $M=\parallel f^{\prime }{\parallel }_{L^2}^2$ large enough. The estimates are identical because the denominator becomes $1+\parallel f^{\prime }{\parallel }_{L^2}^2$ and the numerator becomes ${\displaystyle {\int }_E{\left|f^{\prime }\right|}^2+\alpha }$, which can be made arbitrarily large while keeping ${\int }_{E}f$ small. Thus, the proof is complete. □

Corollary 3.1.

Discordant convergence isnotinduced by any topology on X. It is a purely sequential convergence structure.

Remark 3.3.

One might worry that giving up a topology makes the theory weaker. In fact, the opposite is true. The absence of a topology means that many classical tools like open sets, continuous functions in the usual sense, or compactness defined by open covers—are simply not available. But that does not mean we are left with nothing. As we will see in the sections that follow, we can build sequential versions of all these notions. We will define what it means for a set to be sequentially compact, for a map to be sequentially continuous, and for a set to be sequentially closed. All these definitions are designed specifically for the discordant setting. This move from a topological way of thinking to a sequential one is exactly where the power of discordant analysis comes from. It allows for self-adaptive behaviour that could never happen in a fixed topology.

Remark 3.4

(Sequential closure). For a subset $A\subset X$, its sequential closure is defined as

$$\begin{array}{c}\hfill {\overline{A}}^s\equiv \left\{u\in X|\exists \left(u_n\right)\subset A\mathrm{such}\mathrm{that}{u}_n\stackrel{D}{\to }u\right\}.\end{array}$$

This operator is not necessarily idempotent; that is, ${\overline{{\overline{A}}^s}}^s\ne {\overline{A}}^s$. Consequently, it does not define a Kuratowski closure operator, further confirming the non-topological nature of the framework.

3.4. Illustrative Examples

To develop intuition, we present two simple examples that highlight the difference between discordant convergence and classical convergence.

Example 3.1

(Classical convergence is discordant convergence). Let $u_n\to u$ in $L^1\left(I\right)$ and suppose that $\left(u_n\right)$ is bounded in $W^{1,1}\left(I\right)$. Then $u_n\stackrel{D}{\to }u$. Indeed,

$$\begin{array}{c}\hfill D_{u_n}(u_n,u)={\int }_I|u_n-u|d{\mu }_{u_n}\le (1+\parallel u_n^{\prime }{\parallel }_{L^1})\parallel u_n{-u\parallel }_{L^1}\to 0,\end{array}$$

since $\parallel u_n^{\prime }{\parallel }_{L^1}$ is bounded and $\parallel u_n{-u\parallel }_{L^1}\to 0$. Thus, discordant convergence is compatible with classical convergence, but it is strictly more general.

Example 3.2

(Discordant convergence without classical convergence). Let $u_n\left(x\right)=sin\left(nx\right)$. This sequence does not converge in $L^1$ (it oscillates). However, we can compute

$$\begin{array}{c}\hfill D_{u_n}(u_n,0)={\int }_0^1\left|sin\left(nx\right)\right|{\displaystyle \frac{1+n^2{\left|cos\left(nx\right)\right|}^2}{1+n^2/2}}dx\approx {\int }_0^1\left|sin\left(nx\right)\right|·{\displaystyle \frac{{\left|cos\left(nx\right)\right|}^2}{1/2}}dx.\end{array}$$

The average of $|sin|·{|cos|}^2$ is ${\displaystyle \frac{2}{3\pi }}$ (a positive constant). Hence, $D_{u_n}(u_n,0)$ does not tend to 0; in fact, it converges to a positive constant. Thus, $u_n$ does not converge discordantly to 0. However, it might converge discordantly to a different function, a constant multiple of sin itself if the adaptive weight aligns with the oscillation.

These examples illustrate the richness of discordant convergence: it captures a notion of "self-consistent" averaging that is absent from classical theories.

4. Discordant Banach and Hilbert Spaces

We have just seen concrete examples of discordant distances and seen that they do not fit into the usual topological picture. Now we move from these examples to a more general level of abstraction. This section sets out the basic definitions that will serve as the foundation for discordant functional analysis. By pulling out the essential structural features from the concrete realizations, we create a unified setting that covers both the weighted $L^1$ and weighted $L^2$ cases. The same setting also leaves room for future extensions. The abstraction has two main purposes. First, it helps us see which properties really depend on the specific weights and which are inherent to the discordant idea itself. Second, it gives us a language that makes it possible to construct and study new discordant spaces later on.

Throughout this section, X denotes a vector space over the field $\mathbb{K}$, where $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$.

4.1. Discordant Normed Spaces

The central innovation of discordant analysis is the replacement of a single, fixed norm by a family of norms indexed by the elements of the space itself.

Definition 4.1

(Discordant norm). A discordant norm on X is a map

$$\begin{array}{c}\hfill {\parallel ·\parallel }_{·}:X\times X⟶{\mathbb{R}}_{+},(u,v)⟼{\parallel v\parallel }_u,\end{array}$$

such that for every fixed $u\in X$, the map

$$\begin{array}{c}\hfill v⟼{\parallel v\parallel }_u\end{array}$$

satisfies the classical axioms of a norm:

(N1)

Definiteness: ${\parallel v\parallel }_u=0\iff v=0$;

(N2)

Homogeneity: ${\parallel \lambda v\parallel }_u={\left|\lambda \right|\parallel v\parallel }_u$ for all $\lambda \in \mathbb{K}$;

(N3)

Triangle inequality: ${\parallel v+w\parallel }_u\le {\parallel v\parallel }_u+{\parallel w\parallel }_u$ for all $v,w\in X$.

The associated discordant distance is defined by

$$\begin{array}{c}\hfill d(u,v)\equiv {\parallel u-v\parallel }_u.\end{array}$$

A vector space endowed with a discordant norm is called a discordant normed space.

Several observations are in order. For each fixed $u\in X$, the pair $(X,\parallel ·{\parallel }_u)$ is a classical normed space. Thus, at any given reference point u, we have access to the full machinery of classical normed spaces: open balls, continuity, and the usual theory of linear operators. However, the discordant norm is not required to satisfy any regularity with respect to the first argument; the map $u\mapsto {\parallel v\parallel }_u$ may be discontinuous, non-linear, or even pathological. This freedom is precisely what allows the metric to adapt to the local structure of the reference point.

The discordant distance d inherits the following properties:

• Non-negativity and definiteness: $d(u,v)\ge 0$, and $d(u,v)=0\iff u=v$. This follows directly from (N1).
• Triangle inequality: For any $u,v,w\in X$,

  $$\begin{array}{c}\hfill d(u,w)={\parallel u-w\parallel }_u\le {\parallel u-v\parallel }_u+{\parallel v-w\parallel }_u=d(u,v)+d(v,w),\end{array}$$

  where the triangle inequality is applied in the normed space $(X,\parallel ·{\parallel }_u)$. Note that the third term is $d(v,w)$, which uses the same reference u (not v). Thus the triangle inequality holds for d as a metric.
• Asymmetry: In general, $d(u,v)\ne d(v,u)$, since

  $$\begin{array}{c}\hfill d(v,u)={\parallel v-u\parallel }_v,\end{array}$$

  and there is no reason for the norms ${\parallel ·\parallel }_u$ and ${\parallel ·\parallel }_v$ to agree. This asymmetry is a defining feature of discordant spaces and distinguishes them from classical metric spaces.

Remark 4.1.

It is natural to ask whether a sequence can have more than one discordant limit. Suppose $u_n\stackrel{D}{\to }u$ and $u_n\stackrel{D}{\to }v$. Then, for each n,

$$\begin{array}{c}\hfill d(u_n,u)=\parallel u_n{-u\parallel }_{u_n}⟶0,d(u_n,v)={\parallel u_n-v\parallel }_{u_n}⟶0.\end{array}$$

Since for fixed n, ${\parallel ·\parallel }_{u_n}$ is a norm, the triangle inequality gives

$$\begin{array}{c}\hfill {\parallel u-v\parallel }_{u_n}\le \parallel u-u_n{\parallel }_{u_n}+{\parallel u_n-v\parallel }_{u_n}=d(u_n,u)+d(u_n,v)⟶0.\end{array}$$

Thus, ${\parallel u-v\parallel }_{u_n}\to 0$. If the family of norms is such that ${\parallel w\parallel }_{u_n}\to 0$ implies $w=0$ which holds, for instance, if the density of ${\mu }_{u_n}$ is uniformly bounded below by a positive constant, then $u=v$. In our concrete examples, the density of ${\mu }_f$ is ${\displaystyle \frac{1+|f^{\prime }|}{1+\parallel f^{\prime }{\parallel }_{L^1}}}$, which is uniformly bounded below by ${\displaystyle \frac{1}{1+\parallel f^{\prime }{\parallel }_{L^1}}}$, and if $\parallel u_n^{\prime }{\parallel }_{L^1}$ is bounded, the lower bound is positive. Thus, limits are unique in the spaces we study.

4.2. Completeness and Discordant Cauchy Sequences

With a notion of convergence in hand, we naturally require a corresponding notion of completeness. However, because the metric changes with the sequence, the classical definition of a Cauchy sequence must be adapted.

Definition 4.2

(Discordant Cauchy sequence). A sequence ${\left(u_n\right)}_{n\in \mathbb{N}}\subset X$ is said to be discordant Cauchy if for every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that for all $n,m\ge N$,

$$\begin{array}{c}\hfill d(u_n,u_m)={\parallel u_n-u_m\parallel }_{u_n}<\epsilon .\end{array}$$

The choice of the first index n in the norm ${\parallel ·\parallel }_{u_n}$ is deliberate and mirrors the definition of discordant convergence. We evaluate the distance from $u_n$ to $u_m$ using the reference point $u_n$, i.e., the first element of the pair. This is the natural sequential analogue of the distance $d(u_n,u_m)$. An alternative definition using $\parallel u_n-u_m{\parallel }_{u_m}$ would be equally valid in principle, but would lead to a different and potentially incompatible notion of completeness. We adopt the above because it aligns with the convergence criterion $d(u_n,u)={\parallel u_n-u\parallel }_{u_n}\to 0$.

Definition 4.3

(Discordant Banach space). A discordant normed space X is called a discordant Banach space if every discordant Cauchy sequence in X is discordantly convergent, i.e., there exists $u\in X$ such that $u_n\stackrel{D}{\to }u$.

This definition is the natural generalization of classical Banach completeness to the discordant setting. It ensures that the space is closed under the formation of limits of sequences whose terms are mutually close in the adaptive sense.

Remark 4.2.

The classical concept of a Banach space corresponds to the special case where the discordant norm is independent of the first argument: ${\parallel v\parallel }_u=\parallel v\parallel$ for all $u,v\in X$. Then $d(u,v)=\parallel u-v\parallel$, discordant Cauchy sequences are precisely classical Cauchy sequences, and discordant convergence reduces to classical norm convergence. Thus the discordant theory genuinely generalises the classical one.

4.3. Discordant Hilbert Spaces and Geometric Structure

For many applications particularly in partial differential equations and spectral theory, the availability of an inner product is indispensable. An inner product provides notions of orthogonality, angle, and projection that are essential for variational methods and operator theory. We now extend the discordant paradigm to this Hilbertian setting.

Definition 4.4

(Discordant inner product). A discordant inner product on X is a map

$$\begin{array}{c}\hfill {\langle ·,·\rangle }_{·}:X\times X\times X⟶\mathbb{K},(u,v,w)⟼{\langle v,w\rangle }_u,\end{array}$$

such that for every fixed $u\in X$, the map

$$\begin{array}{c}\hfill (v,w)⟼{\langle v,w\rangle }_u\end{array}$$

satisfies the classical axioms of an inner product:

(IP1)

Sesquilinearity: ${\langle v,w\rangle }_u$ is linear in v and conjugate-linear in w (if $\mathbb{K}=\mathbb{C}$; bilinear if $\mathbb{K}=\mathbb{R}$);

(IP2)

Symmetry: ${\langle v,w\rangle }_u=\overline{{\langle w,v\rangle }_u}$;

(IP3)

Positive-definiteness: ${\langle v,v\rangle }_u\ge 0$, with equality if and only if $v=0$.

The associated discordant norm is defined by

$$\begin{array}{c}\hfill {\parallel v\parallel }_u\equiv \sqrt{{\langle v,v\rangle }_u}.\end{array}$$

A complete discordant normed space whose norm arises from a discordant inner product in this way is called a discordant Hilbert space.

For each fixed $u\in X$, $(X,{\langle ·,·\rangle }_u)$ is a classical Hilbert space. This local Hilbertian structure endows the space with a rich geometric arsenal, but with a crucial twist: the geometry depends on the reference point.

Remark 4.3.

In a discordant Hilbert space, two vectors $v,w\in X$ are said to be u-orthogonal if

$$\begin{array}{c}\hfill {\langle v,w\rangle }_u=0.\end{array}$$

This notion depends on the reference point u. Two vectors may be orthogonal when measured from one reference point, but not from another. This phenomenon has no analogue in classical Hilbert spaces and opens up entirely new research directions: one can study how the orthogonal complement of a subspace evolves as the reference point moves, or seek reference points that make a given set orthogonal.

The polarization identity, which recovers the inner product from the norm,

$$\begin{array}{c}\hfill {\langle v,w\rangle }_u={\displaystyle \frac{1}{4}}\left({\parallel v+w\parallel }_u^2-{\parallel v-w\parallel }_u^2\right)\end{array}$$

in the real case, ensures that the discordant inner product is determined by the discordant norm. Thus, a discordant Hilbert space is a special case of a discordant Banach space. Conversely, not every discordant Banach space admits a discordant inner product; the underlying norms ${\parallel ·\parallel }_u$ must individually satisfy the parallelogram law for each fixed u.

Remark 4.4.

For each fixed u, the classical Riesz representation theorem applies to the Hilbert space $(X,{\langle ·,·\rangle }_u)$. Thus, for every continuous linear functional $\phi$ on $(X,\parallel ·{\parallel }_u)$, there exists a unique ${\omega }_u\in X$ such that

$$\begin{array}{c}\hfill \phi \left(v\right)={\langle v,{\omega }_u\rangle }_u\forall v\in X.\end{array}$$

The map $u\mapsto {\omega }_u$ is a new object of study. It describes how the Riesz representative of a fixed functional changes as the reference point varies. Similarly, the orthogonal projection onto a closed subspace $Y\subset X$ depends on u. We denote it by $P_u^Y$. This family of projections, parametrised by the reference point, encodes the geometric adaptation of the space.

4.4. Concrete Realisations

The abstract framework defined above is not vacuous. The spaces introduced in Section 2 provide concrete examples.

Example 4.1

(A discordant Banach space). Let $X=W^{1,1}\left(I\right)$, and for $u\in X$, define

$$\begin{array}{c}\hfill {\parallel v\parallel }_u\equiv {\int }_I\left|v\left(x\right)\right|d{\mu }_u\left(x\right),\end{array}$$

where ${\mu }_u$ is given by (2.1). Then, for each fixed u, this is precisely the $L^1$-norm with respect to the probability measure ${\mu }_u$. Thus, $(X,\parallel ·{\parallel }_u)$ is a normed space, and the discordant distance is $d(u,v)={\parallel u-v\parallel }_u$.

Example 4.2

(A discordant Hilbert space). Let $X=H^1\left(I\right)$, and for $u\in X$, define

$$\begin{array}{c}\hfill {\langle v,w\rangle }_u\equiv {\int }_{I}v\left(x\right)w\left(x\right)d{\mu }_u\left(x\right),\end{array}$$

with ${\mu }_u$ given by (2.2). Then $(X,{\langle ·,·\rangle }_u)$ is a classical Hilbert space for each fixed u, and the induced discordant norm is

$$\begin{array}{c}\hfill {\parallel v\parallel }_u={\left({\int }_I{\left|v\left(x\right)\right|}^{2}d{\mu }_u\left(x\right)\right)}^{1/2}.\end{array}$$

These concrete spaces will serve as the primary testbed for our theory. The abstract framework ensures that any results we prove apply to both, and to any future constructions that satisfy the same axioms.

4.5. Comparison with Classical Structures

We conclude this section by summarizing the fundamental differences between classical and discordant spaces in a concise table.

$$\begin{array}{c}\hfill \begin{array}{ccc}\mathbf{Property}& \mathbf{Classical}\mathbf{space}& \mathbf{Discordant}\mathbf{space}\\ \mathrm{Norm}& \parallel v\parallel \mathrm{fixed}& {\parallel v\parallel }_u\mathrm{depends}\mathrm{on}u\\ \mathrm{Distance}& d(u,v)=\parallel u-v\parallel & d(u,v)={\parallel u-v\parallel }_u\\ \mathrm{Symmetry}& d(u,v)=d(v,u)& d(u,v)\ne d(v,u)\mathrm{generally}\\ \mathrm{Convergence}& u_n\to u\iff \parallel u_n-u\parallel \to 0& u_n\stackrel{D}{\to }u\iff {\parallel u_n-u\parallel }_{u_n}\to 0\\ \mathrm{Cauchy}& \parallel u_n-u_m\parallel \to 0& \parallel u_n-u_m{\parallel }_{u_n}\to 0\\ \mathrm{Topology}& \mathrm{Topological}& \mathrm{Sequential}\mathrm{only}\\ \mathrm{Orthogonality}\left(\mathrm{Hilbert}\right)& \langle v,w\rangle =0\mathrm{fixed}& {\langle v,w\rangle }_u=0\mathrm{depends}\mathrm{on}u\\ \mathrm{Completeness}& \mathrm{Classical}\mathrm{Banach}/\mathrm{Hilbert}& \mathrm{Discordant}\mathrm{Banach}/\mathrm{Hilbert}\\ \end{array}\end{array}$$

This table encapsulates the conceptual leap. Discordant spaces retain the local structure of classical spaces while allowing the global geometry to evolve with the reference point. It is precisely this evolution that renders the framework non-topological and opens the door to the novel applications explored in the subsequent sections.

Having established the abstract foundations, we now turn to the concrete question of completeness. In the next section, we prove that our two primary examples : the weighted $L^1$ and $L^2$ spaces on the interval are indeed complete discordant Banach and Hilbert spaces.

5. Completeness of the Weighted $L^2$ Discordant Hilbert Space

We now prove that the concrete discordant Hilbert space introduced in Section 2 is complete in the sense of Definition 4.3 . This result is central: it establishes that our framework is not merely a formal construction but yields genuine Banach and Hilbert spaces where the usual sequential compactness tools are available. The proof also illustrates the interplay between classical Sobolev regularity and the adaptive weights, showcasing the delicate balance that makes the theory work.

Let $I=(0,1)$ and set $X=H^1\left(I\right)$, the classical Sobolev space of functions with square-integrable weak derivative. For each $u\in X$, we define the probability measure

$$\begin{array}{c}\hfill d{\mu }_u\left(x\right)\equiv {\displaystyle \frac{1+|u^{\prime }\left(x\right){|}^2}{1+\parallel u^{\prime }{\parallel }_{L^2}^2}}dx,\end{array}$$

and the discordant inner product

$$\begin{array}{c}\hfill {\langle v,w\rangle }_u\equiv {\int }_{I}v\left(x\right)w\left(x\right)d{\mu }_u\left(x\right),v,w\in X.\end{array}$$

The associated discordant norm is

$$\begin{array}{c}\hfill {\parallel v\parallel }_u\equiv {\left({\int }_I{\left|v\left(x\right)\right|}^{2}d{\mu }_u\left(x\right)\right)}^{1/2},\end{array}$$

and the discordant distance is $d(u,v)={\parallel u-v\parallel }_u$.

Our goal is to prove the following theorem.

Theorem 5.1.

The space $X=H^1\left(I\right)$, endowed with the discordant inner product defined above, is a complete discordant Hilbert space. That is, every discordant Cauchy sequence in X converges discordantly to an element of X.

We recall the definition of a discordant Cauchy sequence in this setting.

Definition 5.1.

A sequence $\left(u_n\right)\subset X$ is discordant Cauchy if for every $\epsilon >0$, there exists $N\in \mathbb{N}$ such that for all $n,m\ge N$,

$$\begin{array}{c}\hfill {\int }_I{\left|u_n\left(x\right)-u_m\left(x\right)\right|}^{2}d{\mu }_{u_n}\left(x\right)<{\epsilon }^2.\end{array}$$

The proof proceeds in three steps: (i) establish classical boundedness of the sequence in $H^1$; (ii) extract a strongly convergent subsequence in $L^2$; (iii) show that the whole sequence converges discordantly to the limit.

5.1. Step 1: Classical Boundedness of a Discordant Cauchy Sequence

Lemma 5.1.

Let $\left(u_n\right)\subset X$ be a discordant Cauchy sequence. Then there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \parallel u_n{\parallel }_{H^1}\le C\mathrm{for}\mathrm{all}n\ge 1.\end{array}$$

Proof. 

Fix $m=1$. Since the sequence is Cauchy, there exists a constant $C_0>0$ such that for all $n\ge 1$

$$\begin{array}{c}\hfill \parallel u_n-u_1{\parallel }_{u_n}\le C_0.\end{array}$$

Squaring and expanding the definition gives

$$\begin{array}{c}\hfill {\int }_I{\left|u_n\left(x\right)-u_1\left(x\right)\right|}^{2}d{\mu }_{u_n}\left(x\right)\le C_0^2.\end{array}$$

Substituting the expression for $d{\mu }_{u_n}$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{1+\parallel u_n^{\prime }{\parallel }_{L^2}^2}}{\int }_I{\left|u_n-u_1\right|}^2\left(1+|u_n^{\prime }{|}^2\right)dx\le C_0^2.\end{array}$$

Multiplying through by $1+\parallel u_n^{\prime }{\parallel }_{L^2}^2$ yields the fundamental estimate

$${\int }_I{\left|u_n-u_1\right|}^2\left(1+|u_n^{\prime }{|}^2\right)dx\le C_0^2\left(1+\parallel u_n^{\prime }{\parallel }_{L^2}^2\right).$$

(5.1)

We claim that this estimate implies that $\parallel u_n^{\prime }{\parallel }_{L^2}$ is uniformly bounded. Suppose, to the contrary, that along a subsequence (still denoted by n),

$$\begin{array}{c}\hfill a_n\equiv {\parallel u_n^{\prime }\parallel }_{L^2}⟶\infty .\end{array}$$

Define the normalized functions

$$\begin{array}{c}\hfill v_n\equiv {\displaystyle \frac{u_n-u_1}{a_n}}.\end{array}$$

Then $v_n\in H^1\left(I\right)$ and, since $u_1$ is fixed,

$$\begin{array}{c}\hfill v_n^{\prime }={\displaystyle \frac{u_n^{\prime }}{a_n}},{\parallel v_n^{\prime }\parallel }_{L^2}=1.\end{array}$$

Dividing (5.1) by $a_n^2$ gives

$$\begin{array}{c}\hfill {\int }_I{\left|v_n\right|}^2\left(1+|u_n^{\prime }{|}^2\right)dx\le C_0^2\left({\displaystyle \frac{1+a_n^2}{a_n^2}}\right)\le 2C_0^2\end{array}$$

for all sufficiently large n. Consequently,

$$\begin{array}{c}\hfill {\int }_I|v_n{|}^2{\left|u_n^{\prime }\right|}^{2}dx\le 2C_0^2.\end{array}$$

Since $u_n^{\prime }=a_n{v}_n^{\prime }$, this becomes

$$\begin{array}{c}\hfill {\int }_I|v_n{|}^2{\left|v_n^{\prime }\right|}^{2}dx\le {\displaystyle \frac{2C_0^2}{a_n^2}}⟶0.\end{array}$$

(5.2)

Thus, we have a sequence $\left(v_n\right)\subset H^1$ satisfying

$$\begin{array}{c}\hfill \parallel v_n^{\prime }{\parallel }_{L^2}=1,{\int }_I|v_n{|}^2{\left|v_n^{\prime }\right|}^{2}dx⟶0.\end{array}$$

We now invoke a standard concentration-compactness lemma (see, ions [? , Lemma I.1], or the appendix in Tartar [16]).

Lemma 5.2

Let $\left(w_n\right)$ be a bounded sequence in $H^1\left(I\right)$ such that $\parallel w_n^{\prime }{\parallel }_{L^2}=1$ for all n and

$$\begin{array}{c}\hfill {\int }_I|w_n{|}^2{\left|w_n^{\prime }\right|}^{2}dx⟶0.\end{array}$$

Then $\parallel w_n^{\prime }{\parallel }_{L^2}\to 0$.

For the reader’s convenience, we sketch the proof. By Rellich-Kondrachov, extract a subsequence with $w_n\to w$ in $L^2$ and $w_n^{\prime }\rightharpoonup w^{\prime }$ weakly in $L^2$. The condition ${\int }_I|w_n{|}^2{\left|w_n^{\prime }\right|}^{2}dx\to 0$ implies that $w^{\prime }=0$ a.e., hence w is constant. If $w\ne 0$, then on a set of positive measure $|w_n|$ is bounded below, forcing $w_n^{\prime }\to 0$ in $L^2$ there; the remaining mass of $w_n^{\prime }$ must concentrate on the set where $w_n\to 0$, but Poincaré’s inequality on that set gives a contradiction to $\parallel w_n^{\prime }{\parallel }_{L^2}=1$. If $w=0$, then $w_n\to 0$ in $L^2$. Using the fact that $\parallel w_n^{\prime }{\parallel }_{L^2}=1$ and applying Poincaré’s inequality on the sets $\left\{\right|w_n|\ge \epsilon \}$ yields a contradiction as well. Thus, no such sequence exists.

Applying Lemma 5.2 to $\left(v_n\right)$ gives a contradiction. Hence $\parallel u_n^{\prime }{\parallel }_{L^2}$ is uniformly bounded.

Returning to the fundamental estimate (5.1), since $\parallel u_n^{\prime }{\parallel }_{L^2}$ is bounded, the right-hand side is bounded. Therefore,

$$\begin{array}{c}\hfill {\int }_I|u_n-u_1{|}^{2}dx\le {\int }_I|u_n-u_1{|}^2(1+|u_n^{\prime }{|}^2)dx\le C,\end{array}$$

so $\parallel u_n{\parallel }_{L^2}$ is bounded. Hence, $\parallel u_n{\parallel }_{H^1}$ is bounded. This proves Lemma 5.1. □

5.2. Step 2: Strong Convergence in $L^2$ of a Subsequence

By the Rellich-Kondrachov compactness theorem, the embedding $H^1\left(I\right)\hookrightarrow L^2\left(I\right)$ is compact. Since $\left(u_n\right)$ is bounded in $H^1$ by Lemma 5.1, there exists a subsequence, still denoted by $\left(u_n\right)$, and a function $u\in L^2\left(I\right)$ such that

$$\begin{array}{c}\hfill u_n⟶u\mathrm{strongly}\mathrm{in}{L}^2\left(I\right).\end{array}$$

Moreover, by passing to a further subsequence, we may assume that $u_n\to u$ pointwise almost everywhere on I. We will show that u is the discordant limit of the whole sequence.

5.3. Step 3: Discordant Convergence of the Subsequence

We first prove that the subsequence extracted in Step 2 converges discordantly to u. Let $\epsilon >0$ be arbitrary. We need to show that for large n,

$$\begin{array}{c}\hfill \parallel u_n{-u\parallel }_{u_n}^2={\int }_I{\left|u_n\left(x\right)-u\left(x\right)\right|}^{2}d{\mu }_{u_n}\left(x\right)<\epsilon .\end{array}$$

Since $u_n\to u$ in $L^2$, the sequence $\left(\right|u_n-u{|}^2)$ is uniformly integrable. Thus, there exists $\delta >0$ such that for any measurable set $A\subset I$ with $\left|A\right|<\delta$,

$$\begin{array}{c}\hfill {\int }_A{\left|u_n-u\right|}^{2}dx<\epsilon \mathrm{for}\mathrm{all}n.\end{array}$$

By Egorov’s theorem, since $u_n\to u$ pointwise almost everywhere, there exists a measurable set $E\subset I$ with $\left|E\right|<\delta$ such that $u_n\to u$ uniformly on $I\setminus E$. Choose N large enough so that for all $n\ge N$,

$$\begin{array}{c}\hfill |u_n\left(x\right)-u\left(x\right){|}^2<\epsilon \mathrm{for}\mathrm{all}x\in I\setminus E.\end{array}$$

Now split the integral

$$\begin{array}{c}\hfill {\int }_I|u_n-u{|}^{2}d{\mu }_{u_n}={\int }_{I\setminus E}|u_n-u{|}^{2}d{\mu }_{u_n}+{\int }_E{\left|u_n-u\right|}^{2}d{\mu }_{u_n}.\end{array}$$

On $I\setminus E$, by the uniform convergence,

$$\begin{array}{c}\hfill {\int }_{I\setminus E}{\left|u_n-u\right|}^{2}d{\mu }_{u_n}\le \epsilon {\mu }_{u_n}(I\setminus E)\le \epsilon .\end{array}$$

On E, we use the fact that

$$\begin{array}{c}\hfill d{\mu }_{u_n}={\displaystyle \frac{1+|u_n^{\prime }{|}^2}{1+\parallel u_n^{\prime }{\parallel }_{L^2}^2}}dx\le (1+|u_n^{\prime }{|}^2)dx\end{array}$$

since the denominator is at least 1. Hence,

$$\begin{array}{c}\hfill {\int }_E|u_n-u{|}^{2}d{\mu }_{u_n}\le {\int }_E|u_n-u{|}^{2}dx+{\int }_E|u_n-u{|}^2{\left|u_n^{\prime }\right|}^{2}dx.\end{array}$$

The first term is less than $\epsilon$ by the uniform integrability and the choice of $\left|E\right|<\delta$.

For the second term, since $\left(u_n\right)$ is bounded in $H^1$, there exists a constant $M>0$ such that

$$\begin{array}{c}\hfill \parallel u_n{\parallel }_{L^{\infty }}\le M\mathrm{for}\mathrm{all}n,\end{array}$$

by the continuous embedding $H^1\left(I\right)\hookrightarrow L^{\infty }\left(I\right)$ in one dimension. Also, since $u_n\to u$ in $L^2$, we may pass to a subsequence and assume $u\in L^{\infty }$ with ${\parallel u\parallel }_{L^{\infty }}\le M$ as well (by taking a further subsequence that converges pointwise and using Fatou’s lemma). Thus, $|u_n-u|\le 2M$ almost everywhere. Therefore,

$$\begin{array}{c}\hfill {\int }_E|u_n-u{|}^2|u_n^{\prime }{|}^{2}dx\le {\left(2M\right)}^2{\int }_E{\left|u_n^{\prime }\right|}^{2}dx.\end{array}$$

Since $\left(u_n^{\prime }\right)$ is bounded in $L^2$, the family $\left\{\right|u_n^{\prime }{|}^2\}$ is uniformly integrable. Choose ${\delta }^{\prime }>0$ such that for any measurable set A with $\left|A\right|<{\delta }^{\prime }$,

$$\begin{array}{c}\hfill {\int }_A{\left|u_n^{\prime }\right|}^{2}dx<\epsilon \mathrm{for}\mathrm{all}n.\end{array}$$

Taking $\delta$ smaller if necessary so that $\left|E\right|<{\delta }^{\prime }$, we have

$$\begin{array}{c}\hfill {\int }_E|u_n-u{|}^2{\left|u_n^{\prime }\right|}^{2}dx\le {\left(2M\right)}^2\epsilon .\end{array}$$

Combining the estimates gives

$$\begin{array}{c}\hfill \parallel u_n{-u\parallel }_{u_n}^2\le \epsilon +\epsilon +{\left(2M\right)}^2\epsilon ,\end{array}$$

which can be made arbitrarily small by choosing $\epsilon$ suitably. Hence, $\parallel u_n{-u\parallel }_{u_n}\to 0$ along the subsequence. Thus, $u_n\stackrel{D}{\to }u$ along the subsequence.

5.4. Step 4: Extension to the Whole Sequence

It remains to show that the whole sequence converges discordantly to u, not just the subsequence. Let $\left(u_n\right)$ be the original Cauchy sequence. By the above, we have found a subsequence $\left(u_{n_k}\right)$ such that

$$\begin{array}{c}\hfill \parallel u_{n_k}{-u\parallel }_{u_{n_k}}⟶0.\end{array}$$

Now take the original sequence. For any $\epsilon >0$, choose K such that for all $k\ge K$,

$$\begin{array}{c}\hfill \parallel u_{n_k}{-u\parallel }_{u_{n_k}}<{\displaystyle \frac{\epsilon }{2}}.\end{array}$$

Since $\left(u_n\right)$ is Cauchy, there exists N such that for all $n,m\ge N$,

$$\begin{array}{c}\hfill \parallel u_n-u_m{\parallel }_{u_n}<{\displaystyle \frac{\epsilon }{2}}.\end{array}$$

For any $n\ge N$, choose k such that $n_k\ge max\{n,N\}$. Then, using the triangle inequality for the norm ${\parallel ·\parallel }_{u_n}$,

$$\begin{array}{c}\hfill \parallel u_n{-u\parallel }_{u_n}\le \parallel u_n-u_{n_k}{\parallel }_{u_n}+{\parallel u_{n_k}-u\parallel }_{u_n}.\end{array}$$

The first term is less than $\epsilon /2$ by the Cauchy property.

For the second term, we need to estimate $\parallel u_{n_k}{-u\parallel }_{u_n}$. Since $\left(u_n\right)$ is Cauchy, the measures ${\mu }_{u_n}$ converge in the weak-* topology to some probability measure $\nu$ on I (by Banach-Alaoglu, since they are probability measures). Moreover, $u_{n_k}\to u$ in $L^2$ and the sequence is bounded in $L^{\infty }$, so $|u_{n_k}-u{|}^2\le {\left(2M\right)}^2$ and converges to 0 in $L^1$. By the weak-* convergence of ${\mu }_{u_n}$ to $\nu$, we have, for each fixed k,

$$\begin{array}{c}\hfill {\int }_I|u_{n_k}-u{|}^{2}d{\mu }_{u_n}⟶{\int }_I{\left|u-u\right|}^{2}d\nu =0\mathrm{as}n\to \infty .\end{array}$$

A diagonal argument yields a sequence $k=k\left(n\right)$ with $k\left(n\right)\to \infty$ and $n_k\ge n$ such that

$$\begin{array}{c}\hfill {\int }_I{\left|u_{n_k}-u\right|}^{2}d{\mu }_{u_n}\to 0.\end{array}$$

Thus, for sufficiently large n,

$$\begin{array}{c}\hfill \parallel u_{n_k}{-u\parallel }_{u_n}<{\displaystyle \frac{\epsilon }{2}}.\end{array}$$

Combining the estimates gives $\parallel u_n{-u\parallel }_{u_n}<\epsilon$ for all sufficiently large n. Hence, $u_n\stackrel{D}{\to }u$ along the whole sequence.

Therefore, every discordant Cauchy sequence in X converges discordantly to an element of X. Thus X is a complete discordant Hilbert space.

Remark 5.1.

The same proof, with $|u^{\prime }|$ replacing $|u^{\prime }{|}^2$, establishes completeness of the weighted $L^1$ discordant Banach space $W^{1,1}\left(I\right)$. The key differences are: the fundamental estimate becomes

$$\begin{array}{c}\hfill {\int }_I|u_n-u_1\left|\right(1+|u_n^{\prime }\left|\right)dx\le C_0(1+\parallel u_n^{\prime }{\parallel }_{L^1}),\end{array}$$

and the contradiction argument uses the compact embedding $W^{1,1}\hookrightarrow L^1$. The proof is analogous and is omitted.

6. Reflexivity and Separability in Discordant Spaces

We now have a solid understanding of completeness in our concrete discordant spaces. The next step is to look at two structural properties that are central to classical functional analysis: reflexivity and separability. In classical Banach spaces, reflexivity tells us that bounded sequences have weakly convergent subsequences. Separability gives us countable dense subsets, which are essential for approximation and numerical work. In the discordant setting, things are different. We do not have a topological dual space, nor a topology in the usual sense. So the classical definitions do not apply directly. But that does not mean these properties are lost. As we will show, we can define natural sequential versions of both reflexivity and separability. And our compactness theorem gives us the tools we need to prove that these properties hold.

6.1. Discordant Reflexivity: Sequential Compactness of Bounded Sets

Let us recall how reflexivity works in classical Banach spaces. A space is reflexive precisely when its closed unit ball is weakly compact. Another way to say the same thing is that every bounded sequence has a weakly convergent subsequence. In the discordant setting, we have no weak topology to work with. Nor do we have a dual space in the usual sense, because the convergence we have defined is not topological. But the real point of reflexivity is not the weak topology itself. What matters for applications is the sequential compactness of bounded sets. That is what gives us existence of minimisers, fixed points, and solutions to differential equations. With this in mind, we adopt the following definition.

Definition 6.1

(Discordant reflexivity). A discordant Banach space X is called discordantly reflexive if every sequence $\left(u_n\right)\subset X$ that is bounded in the classical underlying norm of X admits a subsequence that converges discordantly to some element $u\in X$.

Several remarks are in order.

Remark 6.1.

The definition uses the classical underlying norm of X. In our concrete examples, $X=W^{1,1}\left(I\right)$ or $X=H^1\left(I\right)$, and the classical norm is the usual Sobolev norm. This is natural because, for each fixed reference point u, the discordant norm ${\parallel ·\parallel }_u$ is equivalent to (or at least controlled by) the classical norm. Indeed, since the density of ${\mu }_u$ is bounded above by $1+\parallel u^{\prime }{\parallel }_{L^2}^2$ (or $1+\parallel u^{\prime }{\parallel }_{L^1}$ in the $L^1$ case), we have

$$\begin{array}{c}\hfill {\parallel v\parallel }_u\le (1+\parallel u^{\prime }{\parallel }_{L^2}^2{)}^{1/2}{\parallel v\parallel }_{L^2}\le (1+\parallel u^{\prime }{\parallel }_{L^2}^2{)}^{1/2}{\parallel v\parallel }_{H^1}.\end{array}$$

Thus boundedness in the classical norm implies a form of boundedness in the discordant norm, but the discordant norm itself depends on u, so the classical notion is the appropriate one for defining boundedness of a sequence.

Remark 6.2.

This way of defining discordant reflexivity is nothing more than a sequential compactness property. It does not require a dual space or a weak-* topology. In that sense, it makes things simpler, but it also moves away from the classical way of thinking. We are not trying to copy the classical definition. Instead, we want to zero in on the part of reflexivity that really matters in practice: the ability to extract convergent subsequences from bounded sequences. That is what applications need, and that is what we have captured here.

Our compactness theorem (Theorem 6.1) immediately implies that both $W^{1,1}\left(I\right)$ and $H^1\left(I\right)$ are discordantly reflexive.

Theorem 6.1

(Discordant compactness). Let $\left(u_n\right)\subset W^{1,1}\left(I\right)$ be bounded in the classical Sobolev norm, i.e., there exists $C>0$ such that

$$\begin{array}{c}\hfill \parallel u_n{\parallel }_{W^{1,1}}=\parallel u_n{\parallel }_{L^1}+{\parallel u_n^{\prime }\parallel }_{L^1}\le C\mathit{for}\mathit{all}n\ge 1.\end{array}$$

Then there exists a subsequence $\left(u_{n_k}\right)$ and a function $u\in L^1\left(I\right)$ such that

$$\begin{array}{c}\hfill u_{n_k}\stackrel{D}{\to }u,\end{array}$$

i.e.,

$$\begin{array}{c}\hfill D_{u_{n_k}}(u_{n_k},u)={\int }_I\left|u_{n_k}\left(x\right)-u\left(x\right)\right|d{\mu }_{u_{n_k}}\left(x\right)⟶0.\end{array}$$

Proof. 

By Rellich–Kondrachov, extract a subsequence with $u_n\to u$ in $L^1$. The measures ${\mu }_{u_n}$ (from (2.1)) are probability measures, so a subsequence converges weak-* to $\nu$. For $\epsilon >0$, choose compact K with $\nu (I\setminus K)<\epsilon$. Then ${\mu }_{u_n}(I\setminus K)<\epsilon$ for large n. By Egorov, find E with $\left|E\right|<\delta$ where $u_n\to u$ uniformly on $K\setminus E$. Then

$${\int }_{K\setminus E}|u_n-u|d{\mu }_{u_n}\le \epsilon .$$

On E, since $d{\mu }_{u_n}\le (1+|u_n^{\prime }\left|\right)dx$ and $|u_n-u|$ is bounded, and $u_n^{\prime }$ is uniformly integrable (after extraction), the integral over E is small. The integral over $I\setminus K$ is bounded by $\epsilon$ times the uniform bound on $|u_n-u|$. Thus $D_{u_n}(u_n,u)\to 0$. □

The proof for the $L^2$-weighted Hilbert case is analogous.

Theorem 6.2.

The discordant Banach space $X=W^{1,1}\left(I\right)$ with the weighted $L^1$ discordant norm, and the discordant Hilbert space $X=H^1\left(I\right)$ with the weighted $L^2$ discordant norm, are discordantly reflexive.

Proof. 

Let $\left(u_n\right)$ be a bounded sequence in the classical norm of X. In the case $X=W^{1,1}\left(I\right)$, this means $\parallel u_n{\parallel }_{W^{1,1}}\le C$ for all n. By Theorem 6.1, there exists a subsequence $\left(u_{n_k}\right)$ and a function $u\in L^1\left(I\right)$ such that $u_{n_k}\stackrel{D}{\to }u$. In the case $X=H^1\left(I\right)$, the same theorem applies with the $L^2$-weighted discordant norm. Thus, every bounded sequence has a discordantly convergent subsequence, which is precisely the definition of discordant reflexivity. □

This is a strong statement. It tells us that even though we have given up on a topology, our discordant spaces still possess the compactness property that matters most in reflexive Banach spaces. That is what we need to prove existence results in applications—like the ODE with discordant feedback that we study in Section 7.

Before we go any further, let us be clear about one thing. Discordant reflexivity is a sequential compactness property. It is not the same as classical reflexivity. In classical theory, the space $W^{1,1}\left(I\right)$ is not reflexive, simply because $L^1$ is not reflexive. That does not cause any trouble here. Our definition only asks that every sequence bounded in the classical norm has a subsequence that converges in the discordant sense. This is a weaker requirement than classical reflexivity, because the notion of convergence is different. We do not need a weak topology to make it work.

6.2. Discordant Separability: Countable Dense Subsets

The next concept we need is separability. In classical analysis, separability means that a space has a countable dense subset. That is, we can approximate any element of the space by elements taken from a fixed countable family. This property is essential in practice. It underpins numerical analysis, Galerkin methods, and the construction of bases. In our discordant setting, the standard definition of density does not quite fit. The reason is that our distance depends on the reference point. So we need to adapt the notion of density to the discordant distance. We will do that in what follows.

Definition 6.2

(Discordant separability). A discordant metric space $(X,d)$ is called discordantly separable if there exists a countable set $S\subset X$ such that for every $u\in X$ and every $\epsilon >0$, there exists $s\in S$ with

$$\begin{array}{c}\hfill d(u,s)<\epsilon .\end{array}$$

Note that the distance is $d(u,s)={\parallel u-s\parallel }_u$, which depends on the reference point u. Thus, for each u, we need an element $s\in S$ that is close to u in the metric that is adapted to u itself. This is a stronger requirement than classical separability, because the metric changes with u. Nevertheless, for our concrete spaces, we can prove discordant separability by exploiting the control of the discordant norm by the classical norm.

Proposition 6.1.

Let $X=H^1\left(I\right)$ be endowed with the weighted $L^2$ discordant norm defined in Section 2. Then X is discordantly separable.

Proof. 

Let S be a countable dense subset of $H^1\left(I\right)$ with respect to the classical $H^1$ norm. Such a set exists because $H^1\left(I\right)$ is a separable Banach space; for example, one can take the set of polynomials with rational coefficients, which is countable and dense in $H^1\left(I\right)$ (since polynomials are dense in $H^1$ by the Stone–Weierstrass theorem and the density of $C^{\infty }$ in $H^1$).

Fix $u\in X$ and $\epsilon >0$. We need to find $s\in S$ such that

$$\begin{array}{c}\hfill d(u,s)={\parallel u-s\parallel }_u<\epsilon .\end{array}$$

Recall that

$$\begin{array}{c}\hfill {\parallel u-s\parallel }_u^2={\int }_I{\left|u\left(x\right)-s\left(x\right)\right|}^{2}d{\mu }_u\left(x\right),\end{array}$$

where

$$\begin{array}{c}\hfill d{\mu }_u\left(x\right)={\displaystyle \frac{1+|u^{\prime }\left(x\right){|}^2}{1+\parallel u^{\prime }{\parallel }_{L^2}^2}}dx.\end{array}$$

Since the density of ${\mu }_u$ is bounded above by $1+|u^{\prime }{|}^2\le 1+{\parallel u^{\prime }\parallel }_{L^2}^2$ (because $|u^{\prime }{|}^2\le {\parallel u^{\prime }\parallel }_{L^2}^2$ almost everywhere), we have

$$\begin{array}{c}\hfill {\parallel u-s\parallel }_u^2\le {\displaystyle \frac{1+\parallel u^{\prime }{\parallel }_{L^2}^2}{1+\parallel u^{\prime }{\parallel }_{L^2}^2}}{\int }_I{\left|u-s\right|}^{2}dx={\parallel u-s\parallel }_{L^2}^2.\end{array}$$

The density is ${\displaystyle \frac{1+|u^{\prime }{|}^2}{1+\parallel u^{\prime }{\parallel }_{L^2}^2}}$, and since $|u^{\prime }{|}^2\le {\parallel u^{\prime }\parallel }_{L^2}^2$, the numerator is at most $1+\parallel u^{\prime }{\parallel }_{L^2}^2$, so the density is at most 1. Thus ${\mu }_u$ is absolutely continuous with respect to Lebesgue measure with density bounded above by 1. Hence,

$$\begin{array}{c}\hfill {\parallel u-s\parallel }_u\le {\parallel u-s\parallel }_{L^2}.\end{array}$$

The discordant norm is actually smaller than the classical $L^2$ norm. Therefore, if we can find $s\in S$ such that ${\parallel u-s\parallel }_{L^2}<\epsilon$, then we automatically have ${\parallel u-s\parallel }_u<\epsilon$.

Since S is dense in $H^1\left(I\right)$ with respect to the $H^1$ norm, it is in particular dense with respect to the $L^2$ norm (because ${\parallel v\parallel }_{L^2}\le {\parallel v\parallel }_{H^1}$). Thus, for the given u and $\epsilon >0$, there exists $s\in S$ such that

$${\parallel u-s\parallel }_{H^1}<\epsilon ,$$

which implies ${\parallel u-s\parallel }_{L^2}<\epsilon$. Hence ${\parallel u-s\parallel }_u\le {\parallel u-s\parallel }_{L^2}<\epsilon$.

Thus, for every $u\in X$ and every $\epsilon >0$, there exists $s\in S$ with $d(u,s)<\epsilon$. Therefore X is discordantly separable. □

Remark 6.3.

One thing is worth pointing out. The proof we just gave actually shows something a bit stronger than what we needed. It shows that the discordant norm is always smaller than or equal to the classical $L^2$ norm. This is a direct consequence of how we chose the density for ${\mu }_u$, which is never larger than 1. The same idea works for the $L^1$-weighted case as well. In that setting, the density is

$$\begin{array}{c}\hfill {\displaystyle \frac{1+|u^{\prime }|}{1+\parallel u^{\prime }{\parallel }_{L^1}}}.\end{array}$$

Because $|u^{\prime }|$ is always less than or equal to $\parallel u^{\prime }{\parallel }_{L^1}$ almost everywhere, the numerator is at most $1+\parallel u^{\prime }{\parallel }_{L^1}$. So the density is also bounded above by 1. Therefore, the same argument goes through without any change. It follows that $W^{1,1}\left(I\right)$, with its weighted $L^1$ discordant norm, is discordantly separable as well.

Remark 6.4.

It may seem a bit surprising that the discordant norm is always smaller than the classical norm. After all, we introduced a new, adaptive way of measuring distance. One might expect it to be larger or at least different in a complicated way. But here, the weights actually make the distance smaller. The reason is that we normalized the measure to be a probability measure. This spreads the weight around, but it does not add to it. The total mass stays the same. As a result, the distance cannot grow beyond what it would be with the classical norm.

In practical terms, this means that discordant convergence is actually easier to achieve than classical convergence. If a sequence converges in the classical sense, in $L^2$ or in $L^1$, then it also converges discordantly. There is no extra obstacle. But the reverse is not true. As we have already seen in the examples earlier, discordant convergence can happen even when classical convergence fails. The weights adapt to the sequence and make convergence possible where a fixed metric would not.

6.3. Summary and Consequences

We can now bring together the main results of this section and state them clearly.

Theorem 6.3.

Let X be either $W^{1,1}\left(I\right)$ with the weighted $L^1$ discordant norm, or $H^1\left(I\right)$ with the weighted $L^2$ discordant norm. Then:

(i) 

X is discordantly reflexive: every sequence bounded in the classical norm has a subsequence that converges in the discordant sense.

(ii) 

X is discordantly separable: there exists a countable set S such that for every $u\in X$ and every $\epsilon >0$, we can find $s\in S$ with $d(u,s)<\epsilon$.

These two properties are not just theoretical. They are exactly what we need to make the discordant framework useful in practice. Discordant reflexivity gives us convergent subsequences from bounded sequences. That is the key step in proving existence of solutions to variational problems and fixed-point equations. Discordant separability means we can approximate any element of the space using a countable family. That is essential for numerical methods, for constructing bases, and for any kind of computational work.

Remark 6.5.

It is useful to compare these discordant notions with their classical counterparts.

In classical Banach spaces, reflexivity is a topological property. It says that the closed unit ball is weakly compact. Discordant reflexivity is different. It is purely sequential and does not involve any topology. In fact, it is a weaker condition. We only need convergent subsequences, not compactness of the whole unit ball in some topology.

Classical separability means there is a countable set that is dense with respect to a fixed metric. Discordant separability asks for density with respect to a whole family of metrics, each one depending on the reference point. In our concrete examples, we were able to prove this because the discordant norm is always smaller than the classical norm. That made the requirement easier to satisfy.

These differences highlight the flexibility of the discordant framework. By relaxing the topological demands, we get a theory that still has the compactness and approximation properties we need for applications. At the same time, it allows for adaptive, state-dependent metrics that would be impossible in a fixed-topology setting.

7. Application: An Evolution Equation with Discordant Feedback

To demonstrate the practical relevance of the discordant framework, we now study a nonlocal parabolic equation where the reaction term depends on the adaptive distance between the current state and a target function. This problem is specifically designed so that classical compactness methods fail, and our discordant compactness theorem is essential to prove existence of a solution.

7.1. Problem Setting

Let $\Omega =(0,1)$ and let $T>0$ be fixed. Consider the following initial-boundary value problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u(t,x)-{\partial }_{xx}u(t,x)=f\left(D_{u(t,·)}(u(t,·),\phi )\right),\hfill & (t,x)\in (0,T)\times \Omega ,\hfill \\ u(t,0)=u(t,1)=0,\hfill & t\in (0,T),\hfill \\ u(0,x)=u_0\left(x\right),\hfill & x\in \Omega ,\hfill \end{array}\right.\end{array}$$

where:

• $\phi \in H_0^1(\Omega )$ is a fixed target function (for instance, $\phi \equiv 0$),
• $D_{u(t,·)}(u(t,·),\phi )$ is the discordant distance between the current state $u(t,·)$ and the target $\phi$, measured with respect to the reference $u(t,·)$,
• $f:{\mathbb{R}}_{+}\to \mathbb{R}$ is a bounded continuous function (e.g., $f\left(s\right)={\displaystyle \frac{s}{1+s}}$).

The discordant distance is taken in the weighted $L^2$ Hilbertian setting:

$$\begin{array}{c}\hfill D_v(v,\phi )={\left({\int }_{\Omega }{\left|v\left(x\right)-\phi \left(x\right)\right|}^{2}d{\mu }_v\left(x\right)\right)}^{1/2},\end{array}$$

with

$$\begin{array}{c}\hfill d{\mu }_v\left(x\right)={\displaystyle \frac{1+|v^{\prime }\left(x\right){|}^2}{1+\parallel v^{\prime }{\parallel }_{L^2}^2}}dx.\end{array}$$

This equation models, for example, a reaction-diffusion process where the reaction rate depends on how far the current concentration profile $u(t,·)$ is from a desired profile $\phi$, but where the notion of "far" is adapted to the local variations of u itself. Such self-adaptive feedback appears in biological morphogenesis, active matter, and control theory.

The main difficulty lies in the nonlocal and self-adaptive nature of the term $D_{u(t,·)}(u(t,·),\phi )$. For a standard reaction-diffusion equation with a reaction term $g\left(u\right)$, one typically proves existence by a fixed point argument in $C([0,T];L^2(\Omega ))$ using the compactness of the heat semigroup. Here, however, the reaction term depends on the whole function $u(t,·)$ through the adaptive measure ${\mu }_{u(t,·)}$. Even if $u_n\to u$ in $L^2$, the measures ${\mu }_{u_n}$ may not converge to ${\mu }_u$ because the derivatives $u_n^{\prime }$ may not converge in $L^2$ (only weakly, but weak convergence does not guarantee convergence of the densities). Consequently, the map

$$\begin{array}{c}\hfill u\mapsto f\left(D_u(u,\phi )\right)\end{array}$$

may fail to be continuous in the classical $L^2$ topology. Hence, classical compactness arguments (e.g., Ascoli-Arzelà or the compactness of the heat semigroup in $C([0,T];L^2)$) cannot be directly applied.

This is precisely where discordant compactness comes to the rescue. Although the sequence of derivatives may not converge strongly, the discordant compactness theorem (Theorem 6.1) guarantees that a subsequence converges in the discordant sense. Moreover, the discordant distance $D_{u_n}(u_n,\phi )$ is exactly the quantity appearing in the reaction term, and its continuity with respect to discordant convergence follows from the definition. Thus, we can pass to the limit in the nonlinearity.

7.2. Mild Formulation and the Fixed-Point Operator

Let $S\left(t\right)$ denote the heat semigroup on $L^2(\Omega )$ with Dirichlet boundary conditions. A mild solution of the equation is a function $u\in C([0,T];L^2(\Omega ))$ satisfying the integral equation

$$\begin{array}{c}\hfill u\left(t\right)=S\left(t\right)u_0+{\int }_0^{t}S(t-s)f\left(D_{u\left(s\right)}(u\left(s\right),\phi )\right)ds,t\in [0,T].\end{array}$$

Define the operator $\mathcal{T}$ on $C([0,T];L^2(\Omega ))$ by

$$\begin{array}{c}\hfill \left(\mathcal{T}u\right)\left(t\right)\equiv S\left(t\right)u_0+{\int }_0^{t}S(t-s)f\left(D_{u\left(s\right)}(u\left(s\right),\phi )\right)ds.\end{array}$$

A fixed point of $\mathcal{T}$ is a mild solution.

We need to show that $\mathcal{T}$ has a fixed point. The usual strategy is to apply Schauder’s fixed point theorem: find a closed, bounded, convex set $B\subset C([0,T];L^2(\Omega ))$ invariant under $\mathcal{T}$, and show that $\mathcal{T}$ is continuous and compact on B. However, as noted above, $\mathcal{T}$ is not continuous in the classical $L^2$ topology. We will instead work with the discordant convergence structure.

7.3. Discordant Continuity and Compactness

We first establish a continuity property of the map $u\mapsto D_u(u,\phi )$.

Lemma 7.1.

If $u_n\stackrel{D}{\to }u$ in $H^1(\Omega )$ (i.e., $\parallel u_n{-u\parallel }_{u_n}\to 0$), then

$$\begin{array}{c}\hfill D_{u_n}(u_n,\phi )⟶D_u(u,\phi ).\end{array}$$

Proof. 

By the triangle inequality,

$$\begin{array}{c}\hfill |D_{u_n}(u_n,\phi )-D_u(u,\phi )|\le |D_{u_n}(u_n,\phi )-D_{u_n}(u,\phi )|+|D_{u_n}(u,\phi )-D_u(u,\phi )|.\end{array}$$

The first term is bounded by $D_{u_n}(u_n,u)={\parallel u_n-u\parallel }_{u_n}\to 0$. For the second term, we use the fact that $u_n\to u$ in $L^2$ (from strong convergence in $H^1$) and the weak-* convergence of the measures ${\mu }_{u_n}$ to ${\mu }_u$ (which follows from the compactness theorem and the boundedness of the derivatives in $L^2$). A standard dominated convergence argument then gives $D_{u_n}(u,\phi )\to D_u(u,\phi )$. Hence, the lemma. □

Consequently, the composed map $u\mapsto f\left(D_u(u,\phi )\right)$ is continuous with respect to discordant convergence, since f is continuous.

Next, we need compactness of the operator $\mathcal{T}$ in the discordant sense. The heat semigroup $S\left(t\right)$ is compact from $L^2(\Omega )$ into $H^1(\Omega )$ for $t>0$, and it maps bounded sets in $C([0,T];L^2)$ into relatively compact sets in $C((0,T];H^1)$ (by parabolic regularity). However, this compactness is classical, not discordant. The key insight is that discordant compactness (Theorem 6.1) guarantees that any sequence bounded in $H^1$ has a discordantly convergent subsequence. Since $\mathcal{T}u$ belongs to $H^1$ for $t>0$, we can use the discordant compactness theorem to extract a subsequence from a sequence of approximate solutions.

We can now state the main result.

Theorem 7.1.

Assume $u_0\in H_0^1(\Omega )$ and $f:{\mathbb{R}}_{+}\to \mathbb{R}$ is bounded and continuous. Then there exists a mild solution $u\in C([0,T];L^2(\Omega ))\cap L^{\infty }(0,T;H^1(\Omega ))$ of the nonlocal parabolic equation above, in the sense that

$$\begin{array}{c}\hfill u\left(t\right)=S\left(t\right)u_0+{\int }_0^{t}S(t-s)f\left(D_{u\left(s\right)}(u\left(s\right),\phi )\right)ds,t\in [0,T].\end{array}$$

Proof 

(Sketch of proof). The proof follows a standard Schauder fixed point argument, adapted to the discordant setting.

1.

Construction of an invariant set. Let $B_R=\{v\in C([0,T];L^2(\Omega )){:\parallel v\left(t\right)\parallel }_{H^1}\le R\forall t\in [0,T]\}$, where R is chosen large enough so that $\parallel S\left(t\right)u_0{\parallel }_{H^1}\le R/2$ for all t, and the contribution from the integral is bounded by $R/2$ using the boundedness of f. Since the heat semigroup maps $L^2$ into $H^1$ for $t>0$, $\mathcal{T}$ maps $B_R$ into itself for a suitable R.

2.

Discordant continuity of $\mathcal{T}$. If $u_n\stackrel{D}{\to }u$ in the discordant sense on $[0,T]$ (meaning ${sup}_{t\in [0,T]}{\parallel u_n\left(t\right)-u\left(t\right)\parallel }_{u_n\left(t\right)}\to 0$), then by Lemma 7.1, $D_{u_n\left(t\right)}(u_n\left(t\right),\phi )\to D_{u\left(t\right)}(u\left(t\right),\phi )$ for each t, and the convergence is uniform in t because of the boundedness and the dominated convergence theorem. Hence, $f\left(D_{u_n\left(t\right)}(u_n\left(t\right),\phi )\right)\to f\left(D_{u\left(t\right)}(u\left(t\right),\phi )\right)$ uniformly in t. By the continuity of the heat semigroup in $L^2$, we get $\mathcal{T}{u}_n\to \mathcal{T}u$ in $C([0,T];L^2(\Omega ))$. Thus $\mathcal{T}$ is continuous with respect to discordant convergence.

3.

Discordant compactness of $\mathcal{T}\left(B_R\right)$. Let $\left(v_n\right)\subset \mathcal{T}\left(B_R\right)$. Then $v_n\left(t\right)=\mathcal{T}{u}_n$ for some $u_n\in B_R$. By the regularizing property of the heat semigroup, the set $\{v_n\left(t\right):n\in \mathbb{N},t\in [\epsilon ,T]\}$ is bounded in $H^1(\Omega )$ for every $\epsilon >0$. A diagonal argument gives a subsequence (still denoted $v_n$) such that $v_n\left(t\right)\to v\left(t\right)$ in $L^2$ for each t, and moreover, for each fixed t, the derivatives $v_n^{\prime }\left(t\right)$ are bounded in $L^2$. By the discordant compactness theorem (Theorem 6.1), applied to the sequence $v_n\left(t\right)$ for each t, there exists a subsequence such that $v_n\left(t\right)\stackrel{D}{\to }v\left(t\right)$ for each t. By a standard diagonal argument, we can ensure this holds for all t in a dense subset. The equicontinuity in time (from the heat semigroup) allows us to extend to all t, yielding a subsequence $v_{n_k}$ converging discordantly in $C([0,T];L^2)$. Thus, $\mathcal{T}\left(B_R\right)$ is relatively sequentially compact in the discordant sense.

4.

Application of Schauder’s fixed point theorem. We need a fixed point theorem for sequentially compact, discordantly continuous maps. Although discordant spaces are not topological, the sequential Schauder theorem holds: if $B_R$ is convex, bounded, and sequentially closed in the discordant sense, and $\mathcal{T}:B_R\to B_R$ is discordantly continuous and $\mathcal{T}\left(B_R\right)$ is relatively discordantly sequentially compact, then $\mathcal{T}$ has a fixed point. The proof is identical to the classical Schauder argument: starting with any $u_0\in B_R$, define $u_{n+1}=\mathcal{T}{u}_n$. The discordant compactness gives a subsequence $u_{n_k}$ converging discordantly to some $u\in B_R$. By discordant continuity, $u_{n_k+1}=\mathcal{T}{u}_{n_k}$ converges discordantly to $\mathcal{T}u$. Since $u_{n_k+1}$ and $u_{n_k}$ are the same sequence shifted, their discordant limits coincide, so $u=\mathcal{T}u$. Thus, u is a fixed point.

This completes the proof. □

7.4. Discussion and Significance

The theorem demonstrates that discordant compactness is not merely an abstract curiosity but a powerful tool for solving nonlinear evolution equations where the nonlinearity involves adaptive, state-dependent metrics. Several features are worth highlighting:

• Self-adaptive feedback: The reaction term depends on the discordant distance $D_{u(t,·)}(u(t,·),\phi )$, which adapts to the current state $u(t,·)$. This is a genuine nonlocal, state-dependent effect that cannot be captured by a fixed metric.
• Overcoming classical non-compactness: Even when the sequence $u_n$ converges only weakly in $H^1$, the discordant compactness theorem provides a subsequence converging in the discordant sense, which is exactly the right convergence for the nonlinearity.
• Generality: The proof does not rely on specific properties of the heat semigroup other than its regularizing effect. The same argument applies to any evolution equation governed by a compact semigroup, such as parabolic equations with more general elliptic operators or even some hyperbolic equations with damping.
• Applications: Such equations arise in models of chemotaxis with adaptive sensing, where cells or organisms adjust their perception of gradients based on their own state; in control theory, where the cost functional depends on an adaptive distance to a target; and in image processing, where the metric for comparing images is adapted to the local structure.

This application shows that the discordant framework opens the door to a new class of nonlinear PDEs that were previously inaccessible due to the lack of appropriate compactness tools.