Non-Archimedean John, Non-Archimedean Dvoretzky-Milman, Non-Archimedean Type-Cotype and Non-Archimedean Kwapien Problems

1. Non-Archimedean John and Dvoretzky-Milman Problems

Let $\mathcal{X}$ and $\mathcal{Y}$ be two finite dimensional Banach spaces having same dimension. The Banach-Mazur distance between $\mathcal{X}$ and $\mathcal{Y}$ is defined as

$$\begin{array}{c}\hfill d_{BM}(\mathcal{X},\mathcal{Y})inf\{\parallel T\parallel \parallel T^{-1}\parallel :T:\mathcal{X}\to \mathcal{Y}\mathrm{is}\mathrm{invertible}\mathrm{linear}\mathrm{operator}\}.\end{array}$$

(1)

For $n\in \mathbb{N}$, let $({\mathbb{R}}^n,\langle ·,·\rangle )$ be the standard Euclidean space. Famous John’s theorem says following.

Theorem 1. 

[1,2](John Theorem)If $\mathcal{X}$ is any n-dimensional real Banach space, then

$$\begin{array}{c}\hfill d_{BM}(\mathcal{X},({\mathbb{R}}^n,\langle ·,·\rangle ))\le \sqrt{n}.\end{array}$$

Following surprising result says that every finite dimensional Banach space has a subspace which is close (in the Banach-Mazur distance) to Euclidean space [3].

Theorem 2. 

[2,3,4](Dvoretzky-Milman Theorem)There is a universal number $C>0$ satisfying the following property: If $\mathcal{X}$ is any n-dimensional real Banach space and $0<\epsilon <{\displaystyle \frac{1}{3}}$, then for every natural number

$$\begin{array}{c}\hfill k\le C{\displaystyle \frac{{\epsilon }^2}{|log\epsilon |}}logn,\end{array}$$

there exists a k-dimensional subspace $\mathcal{Y}$ of $\mathcal{X}$ such that

$$\begin{array}{c}\hfill d_{BM}(\mathcal{Y},({\mathbb{R}}^k,\langle ·,·\rangle ))<1+\epsilon .\end{array}$$

We refer [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] for more information on Dvoretzky-Milman theorem.

Let $\mathbb{K}$ be a field. A map $|·|:\mathbb{K}\to [0,\infty )$ is said to be a non-Archimedean valuation if following conditions are satisfied.

(i)

If $\lambda \in \mathbb{K}$ is such that $\left|\lambda \right|=0$, then $\lambda =0$.

(ii)

$\left|\lambda \mu \right|=\left|\lambda \right|\left|\mu \right|$ for all $\lambda ,\mu \in \mathbb{K}$.

(iii)

(Ultra-triangle inequality) $|\lambda +\mu |\le max\left\{\right|\lambda |,|\mu \left|\right\}$ for all $\lambda ,\mu \in \mathbb{K}$.

In this case, $\mathbb{K}$ is called as a non-Archimedean valued field [21]. If $\mathbb{K}$ is complete in the ultra-metric

$$\begin{array}{c}\hfill d(\lambda ,\mu )|\lambda -\mu |,\forall \lambda ,\mu \in \mathbb{K},\end{array}$$

then $\mathbb{K}$ is called as a complete non-Archimedean valued field. In the paper, we assume that $\mathbb{K}$ is complete. Let $\mathcal{X}$ be a vector space over a complete non-Archimedean valued field $\mathbb{K}$ with valuation $|·|$. A map $\parallel ·\parallel :\mathcal{X}\to [0,\infty )$ is said to be a non-Archimedean norm if following conditions holds.

(i)

If $x\in \mathcal{X}$ is such that $\parallel x\parallel =0$, then $x=0$.

(ii)

$\parallel \lambda x\parallel =\left|\lambda \right|\parallel x\parallel$ for all $\lambda \in \mathbb{K}$, for all $x\in \mathcal{X}$.

(iii)

(Ultra-norm inequality) $\parallel x+y\parallel \le max\{\parallel x\parallel ,\parallel y\parallel \}$ for all $x,y\in \mathcal{X}$.

In this case, $\mathcal{X}$ is called as a non-Archimedean linear space (NALS) [22]. If $\mathcal{X}$ is complete in the ultra-metric

$$\begin{array}{c}\hfill d(x,y)\parallel x-y\parallel ,\forall x,y\in \mathcal{X},\end{array}$$

then $\mathcal{X}$ is called as a non-Archimedean Banach space. Let $\mathcal{X}$ and $\mathcal{Y}$ be finite dimensional non-Archimedean Banach spaces having same dimension. The non-Archimedean Banach-Mazur distance between $\mathcal{X}$ and $\mathcal{Y}$ is defined as

$$\begin{array}{c}\hfill d_{BM}(\mathcal{X},\mathcal{Y})inf\{\parallel T\parallel \parallel T^{-1}\parallel :T:\mathcal{X}\to \mathcal{Y}\mathrm{is}\mathrm{invertible}\mathrm{linear}\mathrm{operator}\}.\end{array}$$

Given $n\in \mathbb{N}$ and a complete non-Archimedean valued field $\mathbb{K}$, the standard vector space ${\mathbb{K}}^n$ is a non-Archimedean Banach space equipped with norm

$$\begin{array}{c}\hfill \parallel {\left({\lambda }_j\right)}_{j=1}^n\parallel \underset{1\le j\le n}{max}\left|{\lambda }_j\right|,\forall {\left({\lambda }_j\right)}_{j=1}^n\in {\mathbb{K}}^n.\end{array}$$

Problem 1.(Non-Archimedean John Problem)Let $\mathcal{K}$ be the set of all complete non-Archimedean valued fields. What is the best function $\Psi :\mathcal{K}\times \mathbb{N}\to (0,\infty )$ satisfying the following property: If $\mathcal{X}$ is any n-dimensional non-Archimedean Banach space over $\mathbb{K}$, then

$$\begin{array}{c}\hfill d_{BM}(\mathcal{X},({\mathbb{K}}^n,\langle ·,·\rangle ))\le \varphi (\mathbb{K},n).\end{array}$$

Problem 2.(Non-Archimedean Dvoretzky-Mliman Problem)Let $\mathcal{K}$ be the set of all complete non-Archimedean valued fields. What is the best function $\Psi :\mathcal{K}\times (0,\infty )\times \mathbb{N}\to (0,\infty )$ satisfying the following property: If $\mathcal{X}$ is any n-dimensional non-Archimedean Banach space over a complete non-Archimedean valued field $\mathbb{K}$ and $\epsilon >0$, then for every natural number

$$\begin{array}{c}\hfill k\le \Psi (\mathbb{K},\epsilon ,n),\end{array}$$

there exists a k-dimensional complete subspace $\mathcal{Y}$ of $\mathcal{X}$ such that

$$\begin{array}{c}\hfill d_{BM}(\mathcal{Y},({\mathbb{K}}^k,\parallel ·\parallel \left)\right)<1+\epsilon .\end{array}$$

2. Non-Archimedean Type-Cotype and Kwapien Problems

Let $\mathcal{H}$ be a Hilbert space and $n\in \mathbb{N}$. Then for $h_1,\dots ,h_n\in \mathcal{H}$, we have the generalized parallelogram identity

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}{∥\sum _{j=1}^n{\epsilon }_j{h}_j∥}^2=\sum _{j=1}^n{\parallel h_j\parallel }^2.\end{array}$$

(2)

On the other hand, celebrated Jordan-von Neumann result says that Equality (2) characterizes the Hilbert space [23]. Equality (2) motivated the definition of Type and Cotype for Banach spaces.

Definition 1. 

[2] Let $1\le p\le 2$. A Banach space $\mathcal{X}$ is said to be of(Rademacher) Type pif there exists $T_p\left(\mathcal{X}\right)>0$ such that

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}{∥\sum _{j=1}^n{\epsilon }_j{x}_j∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\le T_p\left(\mathcal{X}\right){\left(\sum _{j=1}^n{\parallel x_j\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}},\forall x_1,\dots ,x_n\in \mathcal{X},\forall n\in \mathbb{N}.\end{array}$$

Definition 2. 

[2] Let $2\le q<\infty$. A Banach space $\mathcal{X}$ is said to be of(Rademacher) Cotype qif there exists $C_q\left(\mathcal{X}\right)>0$ such that

$$\begin{array}{c}\hfill {\left(\sum _{j=1}^n{\parallel x_j\parallel }^q\right)}^{{\displaystyle \frac{1}{q}}}\le C_q\left(\mathcal{X}\right){\left({\displaystyle \frac{1}{2^n}}\sum _{{\epsilon }_1,\dots ,{\epsilon }_n\in \{-1,1\}}{∥\sum _{j=1}^n{\epsilon }_j{x}_j∥}^q\right)}^{{\displaystyle \frac{1}{q}}},\forall x_1,\dots ,x_n\in \mathcal{X},\forall n\in \mathbb{N}.\end{array}$$

There is a vast literature on Type-Cotype of Banach spaces, see [2,24,25,26,27,28,29,30,31,32]. In 1972, Kwapien proved the following breakthrough result using Type-Cotype notions [33].

Theorem 3. 

[33,34](Kwapien Theorem)A Banach space $\mathcal{X}$ has Type 2 and Cotype 2 if and only if it is isomorphic to a Hilbert space.

We ask for non-Archimedean version of previous Type-Cotype and Kwapien theorem as follows. Let $\mathbb{K}$ be a complete non-Archimedean valued field. Define

$$\begin{array}{c}\hfill c_0(\mathbb{N},\mathbb{K})\{{\left\{x_n\right\}}_{n=1}^{\infty }:x_n\in \mathbb{K},\forall n\in \mathbb{N},\underset{n\to \infty }{lim}|x_n|=0\}.\end{array}$$

Then $c_0(\mathbb{N},\mathbb{K})$ is a non-Archimedean Banach space w.r.t. the norm

$$\begin{array}{c}\hfill \parallel {\left\{x_n\right\}}_{n=1}^{\infty }\parallel \underset{n\in \mathbb{N}}{max}\left|x_n\right|,\forall {\left\{x_n\right\}}_{n=1}^{\infty }\in c_0(\mathbb{N},\mathbb{K}).\end{array}$$

The space $c_0(\mathbb{N},\mathbb{K})$ is very important (among other non-Archimedean Banach spaces) and is known as p-adic Hilbert space, see [35,36,37].

Problem 3.(Non-Archimedean Type-Cotype and Non-Archimedean Kwapien Problems)Whether there is a way to define non-Archimedean Type and non-Archimedean Cotype for non-Archimedean Banach spaces which will characterize the standard non-Archimedean Banach spaces $c_0(\mathbb{N},\mathbb{K})$? i.e., whether we have a non-Archimedean Kwapien theorem?