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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11
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1answer
287 views

Complemented subspaces in the dual of James' space $J$

James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...
1
vote
1answer
143 views

Hermitian Projections on $C[0,1]$

If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by ...
1
vote
2answers
135 views

Antiproximanal subspace of $L_1[0,1]$

Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$. I read somewhere that $Y$ ...
5
votes
1answer
147 views

Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...
1
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0answers
46 views

Coersivity of a bilinear form [closed]

I need to proof the coersivity of the following bilinear form. a,b and c are scalars, u is the velocity vector field and p is the pressure. Any help is much appreciated! $$ B(\textbf{u},\textbf{v}) = ...
4
votes
1answer
144 views

A quantity measuring weak non-compactness

Let $A$ be a bounded subset of a Banach space $X$. Set: $wk_{X}(A)=\inf\{\epsilon>0:\overline{A}^{w^{*}}\subset X+\epsilon B_{X^{**}}\}$, where $\overline{A}^{w^{*}}$ denotes the $weak^{*}$ closure ...
0
votes
1answer
57 views

The completeness of locally convex space generated by relatively weakly $p$-compact sets

Let $X$ be a Banach space and $1\leq p<\infty$. A bounded subset $K$ of $X$ is relatively weakly $p$-compact if $K$ is contained in $S(B_{l_{p^{*}}})$for some operator $S$ from $l_{p^{*}}$ into ...
4
votes
2answers
188 views

Finitely isometrically persistent metric spaces

The goal of this question is to develop further the discussion initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...
7
votes
1answer
270 views

A Hilbert space characterization via retractions--a conjecture

Given a Banach space $X$ and a functional $f:X\rightarrow \mathbb R$, let $$ X_f := \{x\in X : f(x)\ge 0\} $$ ("functional" means "non-zero linear functional"). Also, given a topological space $E$ ...
9
votes
0answers
220 views

Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?

Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$? It seems to me that it is an interesting ...
6
votes
1answer
297 views

Is $L_q(X^*)$ complemented in $(L_p(X))^*$?

Let $X$ be a Banach space and let $p\in (1,\infty)$. If $q$ denotes the conjugate exponent to $p$, then $L_q(X^*)$ is easily seen to be isometric to a subspace of $(L_p(X))^*$ via the map $$f\mapsto ...
6
votes
1answer
290 views

Embedding of real trees into $\ell_1(\Gamma)$

It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space ...
1
vote
1answer
147 views

Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?

Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space? Clearly if $Y$ is closed in the norm topology ...
14
votes
1answer
370 views

Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof. Question: Let $M$ be a ...
0
votes
0answers
55 views

A question on weakly $p$-convergent sequences

Let $1<p<\infty$. A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is weakly $p$-summable in $X$. We say that a ...
2
votes
1answer
128 views

example of an $\ell_1$-saturated Banach space without an unconditional basis

Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}_p$ which is complementably $\ell_p$-saturated but ...
10
votes
3answers
610 views

Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$

Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
0
votes
0answers
41 views

A weak topology generated by weakly $p$-summable sequences

Let $1\leq p<\infty$ and $X$ be a Banach space. $N_{p}(X)$ is to denote the subspace $\{x^{**}\in X^{**}:$ there exists a weakly $p$-summable sequence $(x_{n})_{n}$ in $X$ such that the sequence ...
2
votes
0answers
145 views

Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold

$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
5
votes
1answer
265 views

Which Banach spaces are realcompact?

I have a question about the topological space underlying a Banach space. A topological space $X$ is realcompact iff it is homeomorphic to a closed subset of an infinite product of the form $\mathbb ...
0
votes
1answer
79 views

On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$ $L^{1}/H^{1}_{0}$ is ...
12
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2answers
371 views

When is the closed unit ball in a smaller Banach space closed in a larger Banach space?

Recently I saw an interesting lemma: For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in ...
0
votes
0answers
94 views

On the operators from $l_{p}$ into Tsirelson's space $T$

Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact? Thank you.
8
votes
3answers
383 views

Thin large subspaces of $\ell^N_1$

Consider a sequence $V_N$ of subspaces of $\ell^N_1$ so that $\dim V_N = N- n$ and $n$ is $\mathsf{o}(N)$. Is it true that these spaces are "thick" (unofficial terminology), i.e. are there constants ...
1
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0answers
32 views

On the normalized weakly $p$-summable sequences in Banach spaces

Let $X$ be a Banach space and $1\leq p<\infty$. My question is: Are the following two statements equivalent: (1). Every normalized weakly $p$-summable sequence in $X$ contains a basic subsequence ...
1
vote
1answer
90 views

A characterization of relatively weakly $p$-compact sets

Let $X$ be a separable Banach space and $1<p<\infty$. We say that a sequence $(x_{n})_{n}$ in $X$ is weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is weakly $p$-summable. A ...
0
votes
1answer
147 views

Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$ that it inherits as the dual of ...
1
vote
1answer
68 views

On the normalized block basic sequences in $c_{0}\widehat{\otimes}_{\pi} c_{0}$

Let $c_{0}\widehat{\otimes}_{\pi} c_{0}$ be the projective tensor product of $c_{0}$ and $c_{0}$. Let $(e_{n})_{n}$ be the unit vector basis of $c_{0}$. For each $n$, define ...
2
votes
1answer
173 views

Equivalent definitions of $\mathscr{L}_p$-spaces

Let $p\in [1,\infty]$ and let $X$ be a separable Banach space. Then $X$ is said to be a $\mathscr{L}_p$-space if there exists an increasing union of finite dimensional Banach spaces $F_n\subset ...
0
votes
1answer
50 views

On the unconditional basis on the Hardy space $H^{1}$ and the Lorentz function space $L_{w,1}$

Question 1. Does the Hardy space $H^{1}$ have an unconditional basis? This problem appeared in S.Kwapien and A.Pelczynski's paper: Some linear topological properties of the hardy spaces $H^{p}$, ...
3
votes
1answer
99 views

Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
1
vote
1answer
87 views

Does the Lorentz spaces $\Lambda(w,1)$ have nontrivial cotype?

I have the following question: does the Lorentz spaces $\Lambda(w,1)$ have nontrivial cotype and admit an unconditional basis(or even a symmetric basis)?Thank you.
1
vote
2answers
201 views

Geodesic on Banach Manifold [closed]

Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?
2
votes
1answer
82 views

Weakly $p$-summable sequences in $L_{r}$

By Bessaga-Pelczynski Selection Principle, it is easy to check that both $l_{p}(1\leq p<2)$ and $l_{r}(1<r<p^{*})$ contains no normalized weakly $p$-summable sequences. I do not know if it is ...
3
votes
1answer
126 views

$M_Λ(A) → A ⊗ M_Λ(C)$

I saw this in here. Let $A$ be a Banach algebra, and let $\Lambda$ be a non-empty set. We denote by $M_\Lambda(A)$ be the set of $\Lambda\times\Lambda$ matrices $(a_{ij})_{i,j\in\Lambda}$ with entries ...
0
votes
0answers
53 views

skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$? For example if $E$ is the space of ...
2
votes
0answers
117 views

Weakly null sequences in $X^{**}/X$

Let $X$ be a space and $Q_{X}:X^{**}\rightarrow X^{**}/X$ be the canonical quotient map. Given a weakly null sequence $(f_{n})_{n}$ in $X^{**}/X$. Is there a weakly Cauchy sequence $(x^{**}_{n})_{n}$ ...
0
votes
2answers
112 views

Weak continuity of a vector valued function [closed]

Let $f:[0,1]\to \ell_\infty[0,1]$ be defined by $f(t)=\chi_{[0,t]}$. Is it true that $f$ is weakly continuous almost everywhere w.r.t. Lebesgue measure ? Here $\ell_\infty[0,1]$ represents the ...
24
votes
0answers
478 views

Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$? There are some results of the ...
1
vote
1answer
99 views

A question about weak convergence on the unit ball of a reflexive space

Which class of reflexive spaces $X$ having the property: if a sequence $(x_{n})_{n}\subset B_{X}$ converges to $x$ weakly and $\|x_{n}\|\rightarrow 1$, then the norm of $x$ must be 1. Of course, the ...
6
votes
1answer
147 views

Is every ideal part of an operator ideal?

An operator ideal $\mathfrak J$ is a class of continuous operators. Namely, for every pair of complex Banach spaces, $\mathfrak X,\mathfrak Y$, we have that $\mathfrak J(\mathfrak X,\mathfrak Y) ...
0
votes
1answer
87 views

On the separability of operator range

Let $T$ be an operator from a Banach space $X$ into a Banach space $Y$ and $1\leq p<\infty$. If $ST$ is compact for any operator $S$ from $Y$ into $l_{p}$, Is $T(X)$ separable? Or under what ...
4
votes
2answers
176 views

Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general: Let $X$ be a separable Banach space and $X^*$ the dual space. The mean value ...
8
votes
0answers
245 views

Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces ...
0
votes
0answers
43 views

On characterizations of $p$-integral operators

In most of papers and classical text books, $p$-integral operators are defined and characterized by operators and commutative diagrams. This is quite different from $p$-summing operators and ...
0
votes
1answer
68 views

Embedding Riemmanian Manifold Linearly

Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?
2
votes
1answer
56 views

A question on unconditionally $p$-summable sequences

We say that a sequence $(x_{n})_{n}$ in a Banach space $X$ is unconditionally $p$-summable ($1\leq p<\infty$) if $$\sup_{x^{*}\in B_{X^{*}}}(\sum_{n=m}^{\infty}|\langle ...
3
votes
0answers
65 views

On the relationship between the factorizations of an operator $T$ and its second adjoint $T^{**}$

Let $T$ be an operator from a space $X$ into a space $Y$ and let $1\leq p<\infty$. If $T^{**}$ has a factorization $T^{**}=RS:X^{**}\xrightarrow{S} l_{p}\xrightarrow{R}Y^{**}$, where $S$ is compact ...
2
votes
1answer
77 views

Post composition of integral

Setup: If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...
1
vote
0answers
68 views

On compact operators with domain $c_{0}$

Let $X$ be a Banach space and $T$ be a compact operator from $c_{0}$ into $X^{*}$. Let $\epsilon>0$. Is there a compact operator $S$ from $X$ into $l_{1}$ such that $S^{*}|_{c_{0}}=T$ and ...