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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Good reference for noncommutative $L^p$ spaces

I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
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0answers
79 views

On the weakly sequential completeness of the dual of the James space $J$

Let me first introduce some definitions. Let $1\leq p\leq \infty$. 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 ...
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2answers
314 views

$l^1$ versus $l^2$

Is there an elementary proof of this Banach space fact? If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...
3
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1answer
167 views

Unconditionally $p$-converging operators on $L_{1}[0,1]$

Let $1\leq p<\infty$. We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-converging if $T$ takes a weakly $p$-summable sequence to a norm null sequence. Question: Is every ...
4
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1answer
231 views

The dual space of $C[0,1]$

Since $C[0,1]^{*}$ is an abstract $L$-space, $C[0,1]^{*}$ is order isometric to $L_{1}(\mu)$ for some measure $\mu$. My question is: the measure $\mu$ can be choosen to be a finite positive measure? ...
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2answers
881 views

Do non-stable Banach spaces exist?

Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties: Is every infinite ...
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0answers
56 views

Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
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66 views

Maurey-Pisier Theorem for complex Banach spaces

A famous theorem of Maurey and Pisier, usually stated for real Banach spaces, says that $\ell_a$ and $\ell_b$ are finitely representable in $X$, where $a$ is the supremum of the $p$ such that $X$ has ...
4
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1answer
322 views

Invariant probability on a unit ball of a Banach space

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries. Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
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1answer
87 views

Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$

Let $X$ be a Banach space with basis $(e_n)_{n=1}^\infty$, and suppose that $(x_i)_{i=1}^\infty$ is a normalized block basic sequence of $(e_n)_{n=1}^\infty$. In addition assume that $(x_i)_{i=1}^\...
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119 views

A Banach space with the BD property and without the weak Gelfand-Phillips property

A subset A of X is called Grothendieck if every operator T from X to $c_0$ maps A to a relatively weakly compact set. A Banach space has the weak Gelfand-Phillips property (wGP) if every ...
4
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1answer
186 views

Non-equivalence of admitting different types of bases in Banach spaces

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...
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3answers
102 views

Classification of subsymmetric basic sequences

I just encountered the following statement: any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell_1$. It's the first time I realize that. I do see the ...
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0answers
43 views

A question on measuring weak non-compactness in $L_{1}(\mu)$ spaces

Let me first fix some notations. Let $X=L_{1}(\mu)$, where $\mu$ is a finite positive measure. Let $A$ be a bounded subset of $X$. Set $\omega(A)=\inf\{\widehat{d}(A,K):\emptyset\neq K\subset X$ is ...
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2answers
161 views

Where can I find some articles and lecture notes in renorming theory in Banach spaces? [closed]

I am really into renorming theory in Banach spaces especially, renorming in non-reflexive Banach spaces such that they have nice property, for example they have fixed point property,locally uniformly ...
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0answers
63 views

Existence of strictly convex Banach space without weak normal structure

This is a complementary question to the following at Existence of normal structure in strictly convex Banach spaces Does there exist a strictly convex Banach space without the weak normal structure ...
4
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1answer
157 views

Is the topological dual of a Banach space weakly* closed in its algebraic dual?

The question is completely contained in the title :) I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...
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272 views

Kolmogorov width for cartesian products

For an operator $T:X\to Y$ between Banach spaces with unit balls $B_X$ and $B_Y$ the sequence of Kolmogorov widths is $$ \delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...
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votes
1answer
99 views

Spectrum of compact operator between different Banach spaces

Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = ...
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530 views

Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...
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2answers
253 views

$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$

$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$. Here,Banach-space isomorphism means a bounded invertible operator ...
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50 views

weak compactness in $(\sum_{n=1}^{\infty}\oplus X_{n})_{p}(1<p<\infty)$

The following result is well-known: Let $(X_{n})_{n}$ be a sequence of Banach spaces and let $X=(\sum_{n=1}^{\infty}\oplus X_{n})_{p}(1<p<\infty)$. Then a bounded subset $A$ of $X$ is ...
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1answer
188 views

Weak*-closure of finite rank operators on dual space

Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is $\overline{F(X^*)...
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53 views

A quantitative version of Pełczyński's property ($V^{*}$)

Let me first fix some notations. Let $A$ be a bounded subset of a Banach space $X$. Set $$wk_{X}(A)=\widehat{d}(\overline{A}^{w^{*}},X)=\sup\{d(x^{**},X):x^{**}\in \overline{A}^{w^{*}}\},$$ where $\...
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87 views

Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$. Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that $...
10
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1answer
241 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 ...
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109 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 ...
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1answer
234 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 $f\in H^s(\mathbb{R}^3)$ ...
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1answer
130 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 ...
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3answers
337 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 ...
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1answer
157 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 f\Vert_{X}^pdt<\...
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1answer
290 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 ...
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1answer
146 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 $V(...
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2answers
137 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
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1answer
158 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 (...
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48 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}) = ...
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1answer
146 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
59 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 $X$....
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2answers
192 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 ...
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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 $E$ ...
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229 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 ...
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votes
1answer
300 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
295 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 $\ell_1(\...
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votes
1answer
82 views

When do two quasi-Banach spaces with identical dual spaces have equivalent norms?

Let $X$ and $Y$ be two quasi-Banach spaces such that the dual spaces satisfy $X^*=Y^*$. I want to know if there are some conditions that imply $X=Y$ (in the sense of equivalent norms).
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154 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 ...
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1answer
401 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 ...
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56 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
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1answer
131 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 ...
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3answers
612 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
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0answers
44 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 $(\...