\documentclass{amsart} % options include 12pt or 11pt or 10pt % classes include article, report, book, letter, thesis \usepackage{setspace} \usepackage{a4} \usepackage{amsthm} \usepackage{latexsym} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{textcomp} \usepackage{cite} \usepackage{enumerate} \usepackage{amssymb} \usepackage{hyperref} \usepackage{amsmath} \usepackage{tikz} \usepackage[mathscr]{euscript} \usepackage{mathtools} \newtheorem{theorem}{Theorem}[section] \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem] {Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{lemma} [theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{observation}[theorem]{Observation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{question}[theorem]{Question} \setlength{\parindent}{0pt} \setlength{\evensidemargin}{0.3cm} \setlength{\oddsidemargin}{0.3cm} \setlength{\topmargin}{-1cm} \textwidth 16cm \textheight 23cm \onehalfspacing \title{This is the title} \newcommand{\Cross}{\mathbin{\tikz [x=1.4ex,y=1.4ex,line width=.2ex] \draw (0,0) -- (1,1) (0,1) -- (1,0);}}% \raggedbottom \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \usepackage{fancyhdr} %\pagestyle{fancy} \pagestyle{fancy} %\thispagestyle{empty} \fancyhead[LO]{\textbf{Non-Archimedean John, Non-Archimedean Dvoretzky-Milman, Non-Archimedean Type-Cotype and Non-Archimedean Kwapien Problems}} \fancyhead[RE]{\textbf{K. MAHESH KRISHNA}} \begin{document} \hrule\hrule\hrule\hrule\hrule \vspace{0.3cm} \begin{center} {\bf{Non-Archimedean John, Non-Archimedean Dvoretzky-Milman, Non-Archimedean Type-Cotype and Non-Archimedean Kwapien Problems}}\\ \vspace{0.3cm} \hrule\hrule\hrule\hrule\hrule \vspace{0.3cm} \textbf{K. MAHESH KRISHNA}\\ School of Mathematics and Natural Sciences\\ Chanakya University Global Campus\\ NH-648, Haraluru Village\\ Devanahalli Taluk, Bengaluru North District\\ Karnataka State, 562 110, India\\ Email: kmaheshak@gmail.com \\ Date: \today \end{center} \hrule\hrule \vspace{0.5cm} %-------------------------------------- \textbf{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. \textbf{Keywords}: John Theorem, Dvoretzky Theorem, Type-Cotype of Banach space, Kwapien Theorem. \textbf{Mathematics Subject Classification (2020)}: 47B10, 47L20, 12J25, 46S10. \hrule %\tableofcontents \hrule \section{Non-Archimedean John and Dvoretzky-Milman Problems} Let $\mathcal{X}$ and $\mathcal{Y}$ be two finite dimensional Banach spaces having same dimension. The \textbf{Banach-Mazur distance} between $\mathcal{X}$ and $\mathcal{Y}$ is defined as \begin{align*} d_{BM}(\mathcal{X}, \mathcal{Y})\coloneqq \inf\{\|T\|\|T^{-1}\|: T: \mathcal{X} \to \mathcal{Y} \text{ is invertible linear operator}\}. \end{align*} For $n \in \mathbb{N}$, let $(\mathbb{R}^n, \langle \cdot, \cdot \rangle)$ be the standard Euclidean space. Famous John's theorem says following. \begin{theorem}\cite{JOHN, ALBIACKALTON} \textbf{(John Theorem)} If $\mathcal{X}$ is any $n$-dimensional real Banach space, then \begin{align*} d_{BM}(\mathcal{X}, (\mathbb{R}^n, \langle \cdot, \cdot \rangle))\leq \sqrt{n}. \end{align*} \end{theorem} Following surprising result says that every finite dimensional Banach space has a subspace which is close (in the Banach-Mazur distance) to Euclidean space \cite{DVORETZKY}. \begin{theorem}\cite{DVORETZKY, MILMAN, ALBIACKALTON} \textbf{(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 $0<\varepsilon < \frac{1}{3}$, then for every natural number \begin{align*} k\leq C \frac{\varepsilon^2}{|\log \varepsilon|}\log n, \end{align*} there exists a $k$-dimensional subspace $\mathcal{Y}$ of $\mathcal{X}$ such that \begin{align*} d_{BM}(\mathcal{Y}, (\mathbb{R}^k, \langle \cdot, \cdot \rangle))<1+\varepsilon. \end{align*} \end{theorem} We refer \cite{MILMANSCHECHTMAN, MILMANBOOK, MILMAN, MILMAN1992, FIGIEL2, SZANKOWSKI, MATOUSEK, MATSAKPLICHKO, PISIERBOOK, GORDONGUEDONMEYER, FIGIEL, FIGIEL3, FIGIELLINDENSTRAUSSMILMAN, SCHECHTMAN, GORDON2, PISIERBOOK2, MILMANSCHECHTMAN, FIGIELLINDENSTRAUSSMILMAN, GORDON3} for more information on Dvoretzky-Milman theorem. Let $\mathbb{K}$ be a field. A map $|\cdot|: \mathbb{K} \to [0, \infty)$ is said to be a \textbf{non-Archimedean valuation} if following conditions are satisfied. \begin{enumerate}[\upshape(i)] \item If $\lambda \in \mathbb{K}$ is such that $|\lambda|=0$, then $\lambda=0$. \item $|\lambda \mu|=|\lambda||\mu|$ for all $\lambda, \mu \in \mathbb{K}$. \item (Ultra-triangle inequality) $|\lambda+\mu|\leq \max\{|\lambda|, |\mu|\}$ for all $\lambda, \mu \in \mathbb{K}$. \end{enumerate} In this case, $\mathbb{K}$ is called as a \textbf{non-Archimedean valued field} \cite{SCHIKHOF}. If $\mathbb{K}$ is complete in the ultra-metric \begin{align*} d(\lambda, \mu)\coloneqq |\lambda-\mu|, \quad \forall \lambda, \mu \in \mathbb{K}, \end{align*} 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 $|\cdot|$. A map $\|\cdot\|: \mathcal{X} \to [0, \infty)$ is said to be a \textbf{non-Archimedean norm} if following conditions holds. \begin{enumerate}[\upshape(i)] \item If $x \in \mathcal{X}$ is such that $\|x\|=0$, then $x=0$. \item $\|\lambda x\|=|\lambda|\|x\|$ for all $\lambda \in \mathbb{K}$, for all $x \in \mathcal{X}$. \item (Ultra-norm inequality) $\|x+y\|\leq \max\{\|x\|, \|y\|\}$ for all $x,y \in \mathcal{X}$. \end{enumerate} In this case, $\mathcal{X}$ is called as a \textbf{non-Archimedean linear space} (NALS) \cite{GARCIASCHIKHOF}. If $\mathcal{X}$ is complete in the ultra-metric \begin{align*} d(x, y)\coloneqq \|x-y\|, \quad \forall x, y \in \mathcal{X}, \end{align*} 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 \textbf{non-Archimedean Banach-Mazur distance} between $\mathcal{X}$ and $\mathcal{Y}$ is defined as \begin{align*} d_{BM}(\mathcal{X}, \mathcal{Y})\coloneqq \inf\{\|T\|\|T^{-1}\|: T: \mathcal{X} \to \mathcal{Y} \text{ is invertible linear operator}\}. \end{align*} 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{align*} \|(\lambda_j)_{j=1}^n \|\coloneqq \max_{1\leq j \leq n}|\lambda_j|, \quad \forall (\lambda_j)_{j=1}^n \in \mathbb{K}^n. \end{align*} \begin{problem} \textbf{(Non-Archimedean John Problem)} Let $\mathscr{K}$ be the set of all complete non-Archimedean valued fields. What is the best function $\Psi: \mathscr{K} \times \mathbb{N} \to (0, \infty)$ satisfying the following property: If $\mathcal{X}$ is any $n$-dimensional non-Archimedean Banach space over $\mathbb{K}$, then \begin{align*} d_{BM}(\mathcal{X}, (\mathbb{K}^n, \langle \cdot, \cdot \rangle))\leq \phi(\mathbb{K}, n). \end{align*} \end{problem} \begin{problem}\label{MODULARDVORETZKYPROBLEM} \textbf{(Non-Archimedean Dvoretzky-Mliman Problem)} Let $\mathscr{K}$ be the set of all complete non-Archimedean valued fields. What is the best function $\Psi: \mathscr{K} \times (0, \infty) \times \mathbb{N} \to (0, \infty)$ 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 $\varepsilon>0$, then for every natural number \begin{align*} k\leq \Psi(\mathbb{K}, \varepsilon, n), \end{align*} there exists a $k$-dimensional complete subspace $\mathcal{Y}$ of $\mathcal{X}$ such that \begin{align*} d_{BM}(\mathcal{Y}, (\mathbb{K}^k, \|\cdot\|))<1+\varepsilon. \end{align*} \end{problem} \section{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{align}\label{H2} \frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} \left\|\sum_{j=1}^{n}\varepsilon_j h_j\right\|^2=\sum_{j=1}^{n}\|h_j\|^2. \end{align} On the other hand, celebrated Jordan-von Neumann result says that Equality (\ref{H2}) characterizes the Hilbert space \cite{JORDANVONNEUMANN}. Equality (\ref{H2}) motivated the definition of Type and Cotype for Banach spaces. \begin{definition}\cite{ALBIACKALTON} Let $1\leq p \leq 2$. A Banach space $\mathcal{X}$ is said to be of \textbf{(Rademacher) Type $p$} if there exists $T_p(\mathcal{X})>0$ such that \begin{align*} \left(\frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} \left\|\sum_{j=1}^{n}\varepsilon_j x_j\right\|^p\right)^\frac{1}{p} \leq T_p(\mathcal{X})\left(\sum_{j=1}^{n}\|x_j\|^p\right)^\frac{1}{p}, \quad \forall x_1, \dots , x_n \in \mathcal{X}, ~\forall n \in \mathbb{N}. \end{align*} \end{definition} \begin{definition}\cite{ALBIACKALTON} Let $2\leq q <\infty$. A Banach space $\mathcal{X}$ is said to be of \textbf{(Rademacher) Cotype $q$} if there exists $C_q(\mathcal{X})>0$ such that \begin{align*} \left(\sum_{j=1}^{n}\|x_j\|^q\right)^\frac{1}{q}\leq C_q(\mathcal{X}) \left(\frac{1}{2^n}\sum_{\varepsilon_1, \dots, \varepsilon_n \in \{-1, 1\}} \left\|\sum_{j=1}^{n}\varepsilon_j x_j\right\|^q\right)^\frac{1}{q}, \quad \forall x_1, \dots , x_n \in \mathcal{X}, ~\forall n \in \mathbb{N}. \end{align*} \end{definition} There is a vast literature on Type-Cotype of Banach spaces, see \cite{LATALAOLESZKIEWICZ, SZAREK, HANDBOOK1, HANDBOOK2, ALBIACKALTON, LIQUEFFELEC1, LIQUEFFELEC2, DIESTELJARCHOWTONGE, MAUREY, PISIERBOOK3}. In 1972, Kwapien proved the following breakthrough result using Type-Cotype notions \cite{KWAPIEN}. \begin{theorem} \cite{KWAPIEN, YAMASAKI} \textbf{(Kwapien Theorem)} A Banach space $\mathcal{X}$ has Type 2 and Cotype 2 if and only if it is isomorphic to a Hilbert space. \end{theorem} 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{align*} c_0(\mathbb{N}, \mathbb{K})\coloneqq \{\{x_n\}_{n=1}^\infty:x_n \in \mathbb{K}, \forall n \in \mathbb{N}, \lim_{n\to \infty}|x_n|=0\}. \end{align*} Then $c_0(\mathbb{N}, \mathbb{K})$ is a non-Archimedean Banach space w.r.t. the norm \begin{align*} \|\{x_n\}_{n=1}^\infty\|\coloneqq \max_{n\in \mathbb{N}}|x_n|, \quad \forall \{x_n\}_{n=1}^\infty\in c_0(\mathbb{N}, \mathbb{K}). \end{align*} 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 \cite{KALISCH, ANIELLOMANCINIPARISI, ANIELLOMANCINIPARISI2}. \begin{problem} \textbf{(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? \end{problem} \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{ALBIACKALTON} Fernando Albiac and Nigel~J. Kalton. \newblock {\em Topics in {B}anach space theory}, volume 233 of {\em Graduate Texts in Mathematics}. \newblock Springer, 2016. \bibitem{ANIELLOMANCINIPARISI2} Paolo Aniello, Stefano Mancini, and Vincenzo Parisi. \newblock Trace class operators and states in {{\(p\)}}-adic quantum mechanics. \newblock {\em J. Math. Phys.}, 64(5):45, 2023. \bibitem{ANIELLOMANCINIPARISI} Paolo Aniello, Stefano Mancini, and Vincenzo Parisi. \newblock Quantum mechanics on a {{\(p\)}}-adic {Hilbert} space: foundations and prospects. \newblock {\em Int. J. Geom. Methods Mod. Phys.}, 21(10):36, 2024. \bibitem{MILMANBOOK} Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali~D. Milman. \newblock {\em Asymptotic geometric analysis. {I}}, volume 202 of {\em Math. Surv. Monogr.} \newblock Providence, RI: American Mathematical Society (AMS), 2015. \bibitem{DIESTELJARCHOWTONGE} Joe Diestel, Hans Jarchow, and Andrew Tonge. \newblock {\em Absolutely summing operators}, volume~43 of {\em Cambridge Studies in Advanced Mathematics}. \newblock Cambridge University Press, Cambridge, 1995. \bibitem{DVORETZKY} Aryeh Dvoretzky. \newblock Some results on convex bodies and {B}anach spaces. \newblock In {\em Proc. {I}nternat. {S}ympos. {L}inear {S}paces ({J}erusalem, 1960)}, pages 123--160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961. \bibitem{FIGIEL3} T.~Figiel. \newblock Some remarks on {D}voretzky's theorem on almost spherical sections of convex bodies. \newblock {\em Colloq. Math.}, 24:241--252, 1971/72. \bibitem{FIGIEL} T.~Figiel. \newblock A short proof of {D}voretzky's theorem. \newblock In {\em S\'{e}minaire {M}aurey-{S}chwartz 1974--1975: {E}spaces {$L^{p}$}, applications radonifiantes et g\'{e}om\'{e}trie des espaces de {B}anach, {E}xp. {N}o. {XXIII}}. 1975. \bibitem{FIGIEL2} T.~Figiel. \newblock A short proof of {D}voretzky's theorem on almost spherical sections of convex bodies. \newblock {\em Compositio Math.}, 33(3):297--301, 1976. \bibitem{FIGIELLINDENSTRAUSSMILMAN} T.~Figiel, J.~Lindenstrauss, and V.~D. Milman. \newblock The dimension of almost spherical sections of convex bodies. \newblock {\em Acta Math.}, 139(1-2):53--94, 1977. \bibitem{GORDONGUEDONMEYER} Y.~Gordon, O.~Guedon, and M.~Meyer. \newblock An isomorphic {D}voretzky's theorem for convex bodies. \newblock {\em Studia Math.}, 127(2):191--200, 1998. \bibitem{GORDON3} Yehoram Gordon. \newblock Some inequalities for {Gaussian} processes and applications. \newblock {\em Isr. J. Math.}, 50:265--289, 1985. \bibitem{GORDON2} Yehoram Gordon. \newblock Gaussian processes and almost spherical sections of convex bodies. \newblock {\em Ann. Probab.}, 16(1):180--188, 1988. \bibitem{JOHN} Fritz John. \newblock Extremum problems with inequalities as subsidiary conditions. \newblock Studies {Essays}, pres. to {R}. {Courant}, 187-204 (1948)., 1948. \bibitem{HANDBOOK1} W.~B. Johnson and J.~Lindenstrauss, editors. \newblock {\em Handbook of the geometry of {B}anach spaces. {V}ol. {I}}. \newblock North-Holland Publishing Co., Amsterdam, 2001. \bibitem{HANDBOOK2} W.~B. Johnson and J.~Lindenstrauss, editors. \newblock {\em Handbook of the geometry of {B}anach spaces. {V}ol. 2}. \newblock North-Holland, Amsterdam, 2003. \bibitem{JORDANVONNEUMANN} P.~Jordan and J.~von Neumann. \newblock On inner products in linear, metric spaces. \newblock {\em Ann. Math. (2)}, 36:719--723, 1935. \bibitem{KALISCH} G.~K. Kalisch. \newblock On p-adic {Hilbert} spaces. \newblock {\em Ann. Math. (2)}, 48:180--192, 1947. \bibitem{KWAPIEN} S.~Kwapie{\'n}. \newblock Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. \newblock {\em Stud. Math.}, 44:583--595, 1972. \bibitem{LATALAOLESZKIEWICZ} Rafal Latala and Krzysztof Oleszkiewicz. \newblock On the best constant in the {K}hinchin-{K}ahane inequality. \newblock {\em Studia Math.}, 109(1):101--104, 1994. \bibitem{LIQUEFFELEC1} Daniel Li and Herve Queffelec. \newblock {\em Introduction to {B}anach spaces: analysis and probability. {V}ol. 1}, volume 166 of {\em Cambridge Studies in Advanced Mathematics}. \newblock Cambridge University Press, Cambridge, 2018. \bibitem{LIQUEFFELEC2} Daniel Li and Herve Queffelec. \newblock {\em Introduction to {B}anach spaces: analysis and probability. {V}ol. 2}, volume 167 of {\em Cambridge Studies in Advanced Mathematics}. \newblock Cambridge University Press, Cambridge, 2018. \bibitem{MATOUSEK} J.~Matousek. \newblock {\em Lectures on discrete geometry}, volume 212 of {\em Graduate Texts in Mathematics}. \newblock Springer-Verlag, New York, 2002. \bibitem{MATSAKPLICHKO} Ivan Matsak and Anatolij Plichko. \newblock Dvoretzky's theorem by {G}aussian method. \newblock In {\em Functional analysis and its applications}, volume 197 of {\em North-Holland Math. Stud.}, pages 171--184. Elsevier Sci. B. V., Amsterdam, 2004. \bibitem{MAUREY} Bernard Maurey. \newblock Type, cotype and {{\(K\)}}-convexity. \newblock In {\em Handbook of the geometry of Banach spaces. Volume 2}, pages 1299--1332. Amsterdam: North-Holland, 2003. \bibitem{MILMAN1992} V.~Milman. \newblock Dvoretzky's theorem---thirty years later. \newblock {\em Geom. Funct. Anal.}, 2(4):455--479, 1992. \bibitem{MILMAN} V.~D. Milman. \newblock A new proof of {A}. {D}voretzky's theorem on cross-sections of convex bodies. \newblock {\em Funkcional. Anal. i Prilo\v{z}en.}, 5(4):28--37, 1971. \bibitem{MILMANSCHECHTMAN} Vitali~D. Milman and Gideon Schechtman. \newblock {\em Asymptotic theory of finite-dimensional normed spaces}, volume 1200 of {\em Lecture Notes in Mathematics}. \newblock Springer-Verlag, Berlin, 1986. \bibitem{GARCIASCHIKHOF} C.~Perez-Garcia and W.~H. Schikhof. \newblock {\em Locally convex spaces over non-{Archimedean} valued fields}, volume 119 of {\em Camb. Stud. Adv. Math.} \newblock Cambridge: Cambridge University Press, 2010. \bibitem{PISIERBOOK3} Gilles Pisier. \newblock {\em Factorization of linear operators and geometry of {Banach} spaces}, volume~60 of {\em Reg. Conf. Ser. Math.} \newblock Providence, RI: American Mathematical Society (AMS), 1986. \bibitem{PISIERBOOK2} Gilles Pisier. \newblock Probabilistic methods in the geometry of {B}anach spaces. \newblock In {\em Probability and analysis ({V}arenna, 1985)}, volume 1206 of {\em Lecture Notes in Math.}, pages 167--241. Springer, Berlin, 1986. \bibitem{PISIERBOOK} Gilles Pisier. \newblock {\em The volume of convex bodies and {B}anach space geometry}, volume~94 of {\em Cambridge Tracts in Mathematics}. \newblock Cambridge University Press, Cambridge, 1989. \bibitem{SCHECHTMAN} Gideon Schechtman. \newblock A remark concerning the dependence on {$\epsilon$} in {D}voretzky's theorem. \newblock In {\em Geometric aspects of functional analysis (1987--88)}, volume 1376 of {\em Lecture Notes in Math.}, pages 274--277. Springer, Berlin, 1989. \bibitem{SCHIKHOF} W.~H. Schikhof. \newblock {\em Ultrametric calculus. {An} introduction to {{\(p\)}}-adic analysis}, volume~4 of {\em Camb. Stud. Adv. Math.} \newblock Cambridge: Cambridge University Press, 2006. \bibitem{SZANKOWSKI} Andrzej Szankowski. \newblock On {D}voretzky's theorem on almost spherical sections of convex bodies. \newblock {\em Israel J. Math.}, 17:325--338, 1974. \bibitem{SZAREK} S.~J. Szarek. \newblock On the best constants in the {K}hinchin inequality. \newblock {\em Studia Math.}, 58(2):197--208, 1976. \bibitem{YAMASAKI} Yasuo Yamasaki. \newblock A simple proof of {Kwapien}'s theorem. \newblock {\em Publ. Res. Inst. Math. Sci.}, 20:1247--1251, 1984. \end{thebibliography} \end{document}