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 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 $(\...
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153 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, ...
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273 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 R^...
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1answer
80 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 ...
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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 $L^...
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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.
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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 $...
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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 $...
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1answer
91 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 ...
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1answer
150 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 $L^{1}(\...
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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 $z_{n}=e_{n}\otimes\sum_{j=...
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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 F_{n+1}...
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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 $H^{p}$, ...
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1answer
139 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 ...
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1answer
89 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.
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207 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?
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83 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 ...
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1answer
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$M_Λ(A) → A ⊗ M_Λ(C)$

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 ...
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57 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 ...
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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 $(x^{**}_{n})_{n}$ ...
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2answers
113 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 ...
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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 ...
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1answer
102 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 ...
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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) \...
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91 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 ...
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2answers
185 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 $...
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254 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 don'...
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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 $p$-...
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1answer
69 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?
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60 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 x^{*},x_{n}\rangle|^{p})^{\...
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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 ...
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1answer
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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 ...
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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 $\|S\|\...
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How to characterize an operator $T$ that factors through a special space?

Let $T\in \mathcal{L}(X,Y)$ and $1<p<\infty$. My question is: Is there a convienent and useful characterization of the operator $T$ factoring through a space $Z$ satisfying $\mathcal{L}(Z,l_{p})=...
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318 views

Two vector spaces with homeomorphic open subsets are isomorphic?

Is it true that if $ E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the ...
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1answer
126 views

An operator factoring through a Banach space containing no copy of $l_{1}$

Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_{1}$, but does not factor through any Banach space $W$ containg no copy of $l_{1}$?...
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1answer
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A question on characterizing a Banach space containing no copy of $l_{1}$

Let $X$ be a Banach space. My question is: $X$ contains no copy of $l_{1}$ if and only if any operator from $X$ to $l_{1}$ is compact? I guess that the necessary part may be true. But is the ...
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1answer
83 views

A question on compact operators with domain $l_{p}$

Suppose that $T$ is an operator from a Banach space $X$ to a Banach space $Y$. Let $1<p<q<\infty$. If $TS$ is compact for any operator $S:l_{p}\rightarrow X$, is $TR$ compact for any operator ...
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Existence of closed operators with arbitrary dense domain of a given banach space

Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$?...
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What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (...
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Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be the corresponding Cameron-Martin Hilbert space (also known as ...
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217 views

does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$. I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...
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The dual of the space of smooth functions that vanish at infinity

Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...
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How to characterize a Banach space $X$ such that any operator from $X$ to $l_{p}$ is compact?

Let $X$ be a Banach space and $1<p<\infty$. How to characterize $X$ such that any operator from $X$ to $l_{p}$ is compact? Are there any known or new results?
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Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing?

A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup_{x^{*}\in B_{X^{*}}}\Bigl(\sum_{n=m}^{\infty}\lvert\langle x^{*},x_{n}\rangle\rvert^{p}\Bigr)^{1/p}\...
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0answers
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Is any operator from $l_{p}$ to a quotient of $l_{r}$($1\leq r<p<\infty$) compact?

Let $1\leq r<p<\infty$. Let $T$ be an operator from $l_{p}$ to a quotient of $l_{r}$. Is $T$ compact?
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“Generalisation” of one-parameter semigroups

Let $(Y,\left\|\cdot\right\|_Y)$ be a Banach space and $A:D(A)\subset Y \to Y$ a closed operator. Studying dynamical systems of the form \begin{equation} u'=Au \end{equation} quickly leads to the ...
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1answer
291 views

Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a $C(...
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2answers
242 views

Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?

In a paper that I am reading there is a following step: Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$. Then $\overline{co}(x_k)$ is a ...
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1answer
275 views

Proof of Pinelis (1992) - Banach space inequalities

I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3: Let $(f_n)$ be a martingale in a separable ...