**9**

votes

**1**answer

154 views

### Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...

**0**

votes

**0**answers

73 views

### A quantity measuring weak compactness

Let us fix some notations. If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $d(A,B)=\inf\{\|a-b\|\mid a\in A,b\in B\},\widehat{d}(A,B)=\sup\{d(a,B)\mid a\in A\}.$ Let $A$ be a ...

**0**

votes

**1**answer

96 views

### Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$
Of course if $f\in H^s(\mathbb{R}^3)$ ...

**1**

vote

**1**answer

74 views

### p-summable sequence

Let Y be a closed linear subspace of X and suppose that Y does not have copy of l1 .Does each weakly p-summable sequence in X/Y has a subsequence that's the image of a weakly p-summable sequence in X ...

**8**

votes

**3**answers

280 views

### Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters

Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...

**1**

vote

**1**answer

136 views

### Dense subspaces of $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert ...

**9**

votes

**0**answers

155 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

**1**answer

134 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

**2**answers

125 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 $Y$ ...

**4**

votes

**1**answer

120 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**

vote

**0**answers

45 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

**1**answer

141 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

**1**answer

51 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

**2**answers

175 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

**1**answer

250 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 $E$ ...

**8**

votes

**0**answers

195 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

**1**answer

282 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

**1**answer

281 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

**1**answer

124 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 ...

**8**

votes

**0**answers

188 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

**0**answers

45 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 ...

**1**

vote

**1**answer

101 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

**3**answers

559 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

**0**answers

31 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

**0**answers

122 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

**1**answer

242 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

**1**answer

71 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**

votes

**2**answers

340 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

**0**answers

92 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

**3**answers

359 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**

vote

**0**answers

24 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

**1**answer

72 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

**1**answer

143 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

**1**answer

56 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

**1**answer

164 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

**1**answer

45 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

**1**answer

96 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

**1**answer

80 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

**2**answers

191 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

**1**answer

68 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

**1**answer

124 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

**0**answers

48 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

**0**answers

110 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

**2**answers

107 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

**0**answers

396 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

**1**answer

89 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

**1**answer

143 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

**1**answer

73 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

**2**answers

159 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

**0**answers

228 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 ...