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

K. MAHESH KRISHNA

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Abstract

We ask for non-Archimedean version of following four: (1) John Theorem, (2) Dvoretzky-Milman Theorem, (3) Type-Cotype of Banach space, (4) Kwapien Theorem.

Non-Archimedean John and Dvoretzky-Milman Problems

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

dBM(X,Y):=inf⁡{∥T∥∥T−1∥:T:X→Y is invertible linear operator}.d_{BM}(\mathcal{X},\mathcal{Y}) := \inf\{\lVert T\rVert\lVert T^{-1}\rVert: T : \mathcal{X} \to\mathcal{Y}\text{ is invertible linear operator}\}.

For n∈Nn \in\mathbb{N}, let (Rn,⟨⋅,⋅⟩)(\mathbb{R}^{n},\langle\cdot,\cdot\rangle) be the standard Euclidean space. Famous John’s theorem says following.

Theorem 1.1. [1, 14] (John Theorem) If X\mathcal{X} is any nn-dimensional real Banach space, then

dBM(X,(Rn,⟨⋅,⋅⟩))≤n.d_{BM}(\mathcal{X},(\mathbb{R}^{n},\langle\cdot,\cdot\rangle)) \le\sqrt{n}.

Following surprising result says that every finite dimensional Banach space has a subspace which is close (in the Banach-Mazur distance) to Euclidean space [6].

Theorem 1.2. [1, 6, 27] (Dvoretzky-Milman Theorem) There is a universal number C>0C > 0 satisfying the following property: If X\mathcal{X} is any nn-dimensional real Banach space and 0<ε<130 < \varepsilon< \frac{1}{3}, then for every natural number

k≤Cε2∣log⁡ε∣log⁡n,k \le C\frac{\varepsilon^{2}}{\lvert\log\varepsilon\rvert}\log n,

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

dBM(Y,(Rk,⟨⋅,⋅⟩))<1+ε.d_{BM}(\mathcal{Y},(\mathbb{R}^{k},\langle\cdot,\cdot\rangle)) < 1+\varepsilon.

We refer [4, 7, 8, 9, 10, 10, 11, 12, 13, 23, 24, 26, 27, 28, 28, 31, 32, 33, 35] for more information on Dvoretzky-Milman theorem.

Let K\mathbb{K} be a field. A map ∣⋅∣:K→[0,∞)\lvert\cdot\rvert: \mathbb{K} \to[0,\infty) is said to be a non-Archimedean valuation if following conditions are satisfied.

(i) If λ∈K\lambda\in\mathbb{K} is such that ∣λ∣=0\lvert\lambda\rvert= 0, then λ=0\lambda= 0.

(ii) ∣λμ∣=∣λ∣∣μ∣\lvert\lambda\mu\rvert= \lvert\lambda\rvert\lvert\mu\rvert for all λ,μ∈K\lambda,\mu\in\mathbb{K}.

(iii) (Ultra-triangle inequality) ∣λ+μ∣≤max⁡{∣λ∣,∣μ∣}\lvert\lambda+\mu\rvert\le\max\{\lvert\lambda\rvert,\lvert\mu\rvert\} for all λ,μ∈K\lambda,\mu\in\mathbb{K}.

In this case, K\mathbb{K} is called as a non-Archimedean valued field [34]. If K\mathbb{K} is complete in the ultra-metric

d(λ,μ):=∣λ−μ∣,∀λ,μ∈K,d(\lambda,\mu) := |\lambda-\mu|,\qquad\forall\lambda,\mu\in\mathbb{K},

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

(i) If x∈Xx\in\mathcal{X} is such that ∥x∥=0\|x\|=0, then x=0x=0.

(ii) ∥λx∥=∣λ∣∥x∥\|\lambda x\|=|\lambda|\|x\| for all λ∈K\lambda\in\mathbb{K}, for all x∈Xx\in\mathcal{X}.

(iii) (Ultra-norm inequality) ∥x+y∥≤max⁡{∥x∥,∥y∥}\|x+y\|\leq\max\{\|x\|,\|y\|\} for all x,y∈Xx,y\in\mathcal{X}.

In this case, X\mathcal{X} is called as a non-Archimedean linear space (NALS) [29]. If X\mathcal{X} is complete in the ultra-metric

d(x,y):=∥x−y∥,∀x,y∈X,d(x,y):=\|x-y\|,\qquad\forall x,y\in\mathcal{X},

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

dBM(X,Y):=inf⁡{∥T∥∥T−1∥:T:X→Y is invertible linear operator}.d_{BM}(\mathcal{X},\mathcal{Y}):=\inf\{\|T\|\|T^{-1}\|:T:\mathcal{X}\to\mathcal{Y}\text{ is invertible linear operator}\}.

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

∥(λj)j=1n∥:=max⁡1≤j≤n∣λj∣,∀(λj)j=1n∈Kn.\left\|\left(\lambda_j\right)_{j=1}^{n}\right\|:=\max_{1\leq j\leq n}|\lambda_j|,\qquad\forall\left(\lambda_j\right)_{j=1}^{n}\in\mathbb{K}^{n}.

Problem 1.3. (Non-Archimedean John Problem) Let K\mathcal{K} be the set of all complete non-Archimedean valued fields. What is the best function Ψ:K×N→(0,∞)\Psi:\mathcal{K}\times\mathbb{N}\to(0,\infty) satisfying the following property: If X\mathcal{X} is any nn-dimensional non-Archimedean Banach space over K\mathbb{K}, then

dBM(X,(Kn,⟨⋅,⋅⟩))≤ϕ(K,n).d_{BM}(\mathcal{X},(\mathbb{K}^{n},\langle\cdot,\cdot\rangle))\leq\phi(\mathbb{K},n).

Problem 1.4. (Non-Archimedean Dvoretzky-Milman Problem) Let K\mathcal{K} be the set of all complete non-Archimedean valued fields. What is the best function Ψ:K×(0,∞)×N→(0,∞)\Psi:\mathcal{K}\times(0,\infty)\times\mathbb{N}\to(0,\infty) satisfying the following property: If X\mathcal{X} is any nn-dimensional non-Archimedean Banach space over a complete non-Archimedean valued field K\mathbb{K} and ε>0\varepsilon>0, then for every natural number

k≤Ψ(K,ε,n),k\leq\Psi(\mathbb{K},\varepsilon,n),

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

dBM(Y,(Kk,∥⋅∥))<1+ε.d_{BM}(\mathcal{Y},(\mathbb{K}^{k},\|\cdot\|))<1+\varepsilon.

Non-Archimedean Type-Cotype and Kwapien Problems

Let H\mathcal{H} be a Hilbert space and n∈Nn\in\mathbb{N}. Then for h1,…,hn∈Hh_{1},\ldots,h_{n}\in\mathcal{H}, we have the generalized parallelogram identity

12n∑ε1,…,εn∈{−1,1}∥∑j=1nεjhj∥2=∑j=1n∥hj∥2.(1)\frac{1}{2^{n}}\sum_{\varepsilon_{1},\ldots,\varepsilon_{n}\in\{-1,1\}}\left\|\sum_{j=1}^{n}\varepsilon_{j}h_{j}\right\|^{2}=\sum_{j=1}^{n}\|h_{j}\|^{2}. \tag*{(1)}

On the other hand, celebrated Jordan-von Neumann result says that Equality (1) characterizes the Hilbert space [17]. Equality (1) motivated the definition of Type and Cotype for Banach spaces.

Definition 2.1. [1] Let 1≤p≤21 \le p \le2. A Banach space X\mathcal{X} is said to be of (Rademacher) Type pp if there exists Tp(X)>0T_p(\mathcal{X}) > 0 such that

(12n∑ε1,…,εn∈{−1,1}∥∑j=1nεjxj∥p)1p≤Tp(X)(∑j=1n∥xj∥p)1p,∀x1,…,xn∈X, ∀n∈N.\left(\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}\left\|\sum_{j=1}^n\varepsilon_jx_j\right\|^p\right)^{\frac{1}{p}}\le T_p(\mathcal{X})\left(\sum_{j=1}^n\|x_j\|^p\right)^{\frac{1}{p}},\quad\forall x_1,\ldots,x_n\in\mathcal{X},\ \forall n\in\mathbb{N}.

Definition 2.2. [1] Let 2≤q<∞2 \le q < \infty. A Banach space X\mathcal{X} is said to be of (Rademacher) Cotype qq if there exists Cq(X)>0C_q(\mathcal{X}) > 0 such that

(∑j=1n∥xj∥q)1q≤Cq(X)(12n∑ε1,…,εn∈{−1,1}∥∑j=1nεjxj∥q)1q,∀x1,…,xn∈X, ∀n∈N.\left(\sum_{j=1}^n\|x_j\|^q\right)^{\frac{1}{q}}\le C_q(\mathcal{X})\left(\frac{1}{2^n}\sum_{\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}}\left\|\sum_{j=1}^n\varepsilon_jx_j\right\|^q\right)^{\frac{1}{q}},\quad\forall x_1,\ldots,x_n\in\mathcal{X},\ \forall n\in\mathbb{N}.

There is a vast literature on Type-Cotype of Banach spaces, see [1, 5, 15, 16, 20, 21, 22, 25, 30, 36]. In 1972, Kwapien proved the following breakthrough result using Type-Cotype notions [19].

Theorem 2.3. [19, 37] (Kwapien Theorem) A Banach space X\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 K\mathbb{K} be a complete non-Archimedean valued field. Define

c0(N,K):={{xn}n=1∞:xn∈K, ∀n∈N, lim⁡n→∞∣xn∣=0}.c_0(\mathbb{N},\mathbb{K}) := \left\{\{x_n\}_{n=1}^{\infty}:x_n\in\mathbb{K},\ \forall n\in\mathbb{N},\ \lim_{n\to\infty}|x_n|=0\right\}.

Then c0(N,K)c_0(\mathbb{N},\mathbb{K}) is a non-Archimedean Banach space w.r.t. the norm

∥{xn}n=1∞∥:=max⁡n∈N∣xn∣,∀{xn}n=1∞∈c0(N,K).\left\|\{x_n\}_{n=1}^{\infty}\right\| := \max_{n\in\mathbb{N}}|x_n|,\quad\forall\{x_n\}_{n=1}^{\infty}\in c_0(\mathbb{N},\mathbb{K}).

The space c0(N,K)c_0(\mathbb{N},\mathbb{K}) is very important (among other non-Archimedean Banach spaces) and is known as p-adic Hilbert space, see [2, 3, 18].

Problem 2.4. (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 c0(N,K)c_0(\mathbb{N},\mathbb{K})? i.e., whether we have a non-Archimedean Kwapien theorem?

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