Questions tagged [banach-spaces]

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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A measure of non-reflexivity of Banach spaces

Let $X$ be a Banach space and define $\gamma(X)=\sup\{|\lim\limits_{n}\lim\limits_{m}\langle x^{*}_{m},x_{n}\rangle-\lim\limits_{m}\lim\limits_{n}\langle x^{*}_{m},x_{n}\rangle|:(x_{n})_{n}$ is a ...
Dongyang Chen's user avatar
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Closure of BV paths in space of paths of finite $p$-variation

Let $p\ge1$ and $T>0$. Define $\mathscr D([0,T])$to be the space of partitions of $[0,T]$, where each partition is a finite collection of distinct points of $[0,T]$. Consider a continuous path $X:[...
Martin Geller's user avatar
1 vote
2 answers
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A bimonotone basis for $\mathcal{C}[0,1]$?

It is well-known that $\mathcal{C}[0,1]$, the space consisting of all scalar-valued continuous functions over the unit interval, has a monotone Schauder basis. In fact, we can construct such a basis ...
Anso's user avatar
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2 answers
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Duality of projective and injective tensor product

I want a reference of the following statement which I think is true. Let $X$ and $Y$ be Banach spaces with $X$ finite dimensional. Then $(X\otimes_\epsilon Y)^*$ is isometrically isomorphic to $(X^*\...
A beginner mathmatician's user avatar
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113 views

Subspaces of inductive limit of Banach spaces

Let $(E_n)_{n\geq 1}$ be a sequence of Banach spaces such that $E_n\subseteq E_{n+1}.$ Assume that $\text{dim}E_n=n.$ Assume that $E$ is a Banach space such that $E_n\subseteq E$ and $\cup_{n\geq 1}...
A beginner mathmatician's user avatar
5 votes
1 answer
183 views

Large ideally convex sets

Let $E$ be a Banach space. A set $C \subseteq E$ is called ideally convex if for every bounded sequence $(x_n)$ in $C$ and for every sequence $(\lambda_n)$ in $[0,1]$ that sums up to $1$ the vector $\...
Jochen Glueck's user avatar
3 votes
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Automatic complete boundedness for bilinear and multilinear maps

$\newcommand{\cb}{\mathrm{cb}}$Let $T : X \rightarrow Y$ be a bounded linear map between Banach spaces. We have the following results concerning automatic complete boundedness: $\|T : X \rightarrow \...
Seven9's user avatar
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A list of properties of $(\bigoplus \ell^1_n)_{\ell^p}$, $1<p<\infty$

The Banach space $E=(\bigoplus_{n=1}^{\infty}\ell^1_n)_{\ell^p}$ for $1<p<\infty$ shows up in various places in the literature to construct counterexamples. The purpose of this post is to ...
Onur Oktay's user avatar
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Bounds on dimension of a subspace

Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that: $$ \| u\|_{...
Ali's user avatar
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The ultrapower of the direct sum is the direct sum of ultrapowers

Currently, I'm reading the paper "Towards the fixed point property for superreflexive spaces" by Andrzej Wiśnicki. In this article, given $X_1,\dots,X_n$ Banach spaces, he defines $(X_1\...
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1 vote
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Representations of the dual Banach algebra pair $(\ell_1,c_0)$

Let $\displaystyle E_p=(\bigoplus_{n\in\mathbb{N}} \ell^1_n)_{\ell^p}$ for some $1<p<\infty$ and $\ell^1 = \ell^1(\mathbb{N})$ be equipped with the convolution. Then, there exists an isometric &...
Onur Oktay's user avatar
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2 votes
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Quotients of $c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1$

Let $\hat{\otimes}_{\pi}$ denote the projective tensor product. Let $$\mathcal{S} = \{V\subseteq c_0\mathbin{\hat{\otimes}_{\pi}}\ell^1\textrm{ closed subspace}: {c_0\mathbin{\hat{\otimes}_{\pi}}\ell^...
Onur Oktay's user avatar
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1 vote
1 answer
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Complex interpolation of subspaces

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\...
A beginner mathmatician's user avatar
2 votes
1 answer
343 views

For a Banach space $X$, can we find a reflexive (or weakly sequentially complete) space $Y$ such that $X\subset Y$?

It could be a naive question. Probably, it is not true. However, this question makes sense in the setting of function spaces. For example, for $L_\infty (0,1)$, we have $L_p(0,1)\supset L_\infty (0,1)$...
user92646's user avatar
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1 answer
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Weak compactness of the closed unit ball of $L_{\infty}(\mu,X)$ in $L_{1}(\mu,X)$

It is known that the closed unit ball of $L_{\infty}(\mu)$ is weakly compact in $L_{1}(\mu)$. A natural question arises in the case of spaces of Bochner integral functions: Question. Let $X$ be a ...
Dongyang Chen's user avatar
1 vote
0 answers
165 views

About a weak$^*$ convergent net

Let $G$ be a locally compact abelian group and $A$ be semisimple commutative Banach algebra such that $A^{**}$ has Radon-Nikodym property. Denote by $\Gamma$ and $M(G)$ the dual group and the measure ...
Meisam Soleimani Malekan's user avatar
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0 answers
163 views

Completely continuous maps from projective tensor products into $c_0$

Let $E$, $F$ be two Banach spaces and $E\mathbin{\hat{\otimes}}_{\pi}F$ denote their projective tensor product. For each unit norm $\xi\in E$ and $\gamma\in F$, let's define $$ J_{\gamma}:E\to E\...
Onur Oktay's user avatar
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1 vote
1 answer
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The space of linear operators between Hilbert spaces has martingale type 2

I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded ...
Shashi's user avatar
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2 votes
1 answer
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$\ell^1$ predual with no $c_0$ quotient?

Question: Does there exist an isomorphic predual of $\ell^1$, which does not have a quotient isomorphic to $c_0$? Thanks in advance. Edit: The answer is no. Let $X$ be a Banach space such that $X^*$...
Onur Oktay's user avatar
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1 vote
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Banach spaces in which every DP-set is a limited set

Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset. $A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \lim_{n\to\...
Onur Oktay's user avatar
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2 votes
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342 views

Weakly null sequences in projective tensor products

First, I'd like to record a question that may still be open. The snippet below is taken from DiestelPuglisi2009. Second, let $E$ be a Banach space, $(u_n)$ be a weakly null sequence in the projective ...
Onur Oktay's user avatar
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1 vote
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Weak convergence using tensor product

I haven't got to see this argument used in the PhD thesis of [R. Ryan]: Applications of topological tensor products to infinite dimensional holomorphy, doctoral thesis, Trinity College, Dublin (1980), ...
Nicolay Avendaño's user avatar
3 votes
0 answers
291 views

Dunford-Pettis like properties for Banach spaces of operators

Let $E$ be a Banach space and $A\subseteq B(E)$ be a Banach subspace of operators on $E$. Suppose $A$ satisfies the property (RCC) given below: $$ \left.\begin{array}{l} (x_n)\subseteq A \textrm{ ...
Onur Oktay's user avatar
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0 votes
3 answers
142 views

Absolutely summing operators from $l_{p}$ to $l_{q}$

Recall that an operator $T:X\rightarrow Y$ is called absolutely summing if there exists a constant $C>0$ such that $$\sum_{i=1}^{n}\|Tx_{i}\|\leq C \sup_{x^{*}\in B_{X^{*}}}\sum_{i=1}^{n}|\langle x^...
Dongyang Chen's user avatar
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91 views

Finding set of best approximations from a point in $c_0$ to its subspace

Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
PPB's user avatar
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2 votes
2 answers
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Schauder bases in Banach spaces with a symmetric $k$-FDD

The Kalton-Peck Banach space $Z_2$ (see Section 6 in this paper) does not admit an unconditional basis, but it admits an unconditional, even symmetric, FDD (finite dimensional decomposition) into ...
M.González's user avatar
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4 votes
0 answers
238 views

Automorphic Banach spaces

A Banach space $X$ is called automorphic if for every closed subspace $Y\subseteq X$ with $\dim X/Y=\infty$, every automorphism (= linear continuous isomorphism) of $Y$ can be extended to an ...
Lviv Scottish Book's user avatar
16 votes
6 answers
2k views

Finding closed subspaces whose sum isn't closed

Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
Nik Weaver's user avatar
5 votes
0 answers
188 views

Given a totally ordered system of Banach spaces, can we we always change the norms to get isometric embeddings?

Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...
Cosine's user avatar
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Gateaux differentiability

I have a question about the Gateaux differentiability of a map on a particular space. To be more specific, let $f: U \to X$ be a map where both $U$ and $X$ are Hilbert spaces. Now, assume that there ...
TOT's user avatar
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2 votes
1 answer
275 views

An abstract characterisation of weak* topologies

Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if ...
HardyHulley's user avatar
3 votes
0 answers
67 views

Making a space UMD via interpolation

Recall that a Banach space $B$ has Unconditional Martingale Difference (UMD-$p$) if there is a constant $C_p$ such that for every $B$-valued martingale difference sequences $(d_n)_n$ and choice of $\...
Marco's user avatar
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1 vote
1 answer
88 views

Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?

Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...
Akira's user avatar
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1 vote
1 answer
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Bilipschitz embedding of the unit ball of $c_0$ into $\ell_1$

This is a follow-up to this question of mine. It is well-known that the Banach space $\ell_1$ does not contain any isomorphic copies of $c_0$. One can even go further and show that $\ell_1$ does not ...
Damian Sobota's user avatar
2 votes
0 answers
125 views

What are examples of infinite-dimensional Banach spaces that are also measure spaces?

I am interested in examples of infinite-dimensional vector spaces that are Banach spaces or even Hilbert spaces are measure spaces Instead of the full vector space, subsets with measure structure ...
shuhalo's user avatar
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1 vote
0 answers
70 views

Lipschitz isomorphisms of $C(\omega^\omega+)$

Let $C(\omega^\omega+)$ denote the Banach space of continuous, scalar-valued functions defined on $\omega^\omega+=[0,\omega^\omega]$. Suppose that $X$ is a Banach space and $U:C(\omega^\omega+)\to X$ ...
user469053's user avatar
7 votes
0 answers
129 views

A geometric intuition about convexifiability

I've come up with the following conjecture about convexifiability being determined by "important" sets in Banach spaces. To me, the conjecture looks quite innocuous and intuitive, but I'm ...
user469053's user avatar
1 vote
1 answer
82 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

This is a follow-up to this previous question, but under stronger assumptions. Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real scalar field). Let $\tilde ...
Jochen Glueck's user avatar
0 votes
1 answer
75 views

On the measure of nonconvexity (MNC)

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by: $$\alpha(A)=\sup_{b\in\operatorname{conv}(A)} \...
Motaka's user avatar
  • 291
0 votes
1 answer
103 views

If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e

I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral. Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \...
Akira's user avatar
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1 vote
1 answer
204 views

Finding the set of best approximation

Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$. Note: A ...
PPB's user avatar
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9 votes
1 answer
633 views

Banach space with uncountable basis

We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach ...
Anupam's user avatar
  • 479
7 votes
1 answer
711 views

Converse of closed graph theorem

Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach ...
Anupam's user avatar
  • 479
0 votes
1 answer
243 views

Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?

I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach ...
Akira's user avatar
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1 vote
1 answer
164 views

Complemented subspaces in a dual Banach space

Let $Y$ be a complemented subspace in a dual Banach space $X$. Is it true that $Y$ is itself isomorphic to a dual? This is the case of a $w^*$-closed subspace $Y$, but a complemented subspace of $X^*...
Pietro Majer's user avatar
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0 votes
1 answer
174 views

A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space

Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L_p := \mathcal L_p (X, \mu, E)$ and $\|\cdot\|_{\mathcal L_p}$ ...
Analyst's user avatar
  • 637
1 vote
1 answer
226 views

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
Analyst's user avatar
  • 637
7 votes
1 answer
217 views

A characterization of Hilbert spaces by norm one projections

Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
Markus's user avatar
  • 1,361
2 votes
1 answer
93 views

Differentiability of the fixed points of a family of contraction maps

Given a general Banach space $B$ and a one-parameter family of contractions $F_t:B\to B$ which is defined for all $t \in (a,b)$. $F_t$ depends continuously on $t$ (in the sense $\lim_{\varepsilon\to 0}...
toaster's user avatar
  • 121
2 votes
0 answers
52 views

The initial sigma-algebra on the dual of a Banach lattice

Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
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