All Questions
1,222 questions
1
vote
0
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80
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Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$
This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
- 2,830
1
vote
2
answers
311
views
Is there a bounded sequence $(e_n)$ such that $e_n \in E_n$ and that $(e_n)$ does not have any convergent subsequence?
Let $(E, |\cdot|)$ be an infinite-dimensional Banach space. Assume that
$T:E\to E$ is a compact (bounded linear) operator, and
$(\lambda_n)$ is a sequence of distinct eigenvalues of $T$.
Let $E_n$ ...
- 657
0
votes
1
answer
165
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For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
- 825
0
votes
0
answers
145
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$L_\infty([0,1], \mathbb{C})$ is it isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{C})$?
By a result of Pełczyński, $L_\infty([0,1], \mathbb{R})$ is isomorphic to $\ell_\infty(\mathbb{N}, \mathbb{R})$. That is the case of real valued functions and sequences.
A natural question then is: ...
- 141
2
votes
2
answers
167
views
LF or LB space that happens to be finite dimensional
Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further ...
- 3,477
4
votes
1
answer
259
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The real and the imaginary part of a vector
In an infinite-dimensional Banah space $(X, \|\cdot\|)$ with a countable Schauder basis $\{x_n\}$, define:
$$
F_r: \operatorname{Span}(\{x_n\}) \rightarrow \operatorname{Span}(\{x_n\}), \hspace{0.3cm} ...
- 307
2
votes
2
answers
160
views
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 ...
- 4,461
7
votes
1
answer
737
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 ...
- 585
2
votes
0
answers
98
views
Geometric interpretation of uniform convexity condition
I first want to recall the moduli of uniform smoothness (US), uniform convexity (UC), asymptotic uniform smoothness (AUS), and asymptotic uniform convexity (AUC). Throughout, let $X$ be an infinite ...
- 41
0
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0
answers
137
views
Convexity of an equivalent norm
Let $X=l_2$ with usual norm $\|\cdot\|_2$. We define a subspace of $X$ as $D=conv (B_{l_2} \cup B),$ where $B = \{ (x_n) \in l_2 : \sum_{n=1}^\infty \frac{n}{2} x_n^2 \leq 1\}$, conv is the convex ...
- 85
2
votes
1
answer
323
views
Weakly compact operators into $c_0$ and other separable spaces
A Banach space $E$ is called Grothendieck if every weak* convergent sequence in the dual space $E^*$ is weakly convergent. A typical example of a Grothendieck space is $\ell_\infty$. Diestel proved ...
- 1,151
2
votes
2
answers
297
views
When a quasinilpotent is nilpotent?
In the case of an infinite-dimensional complex banach space $X$,
under what conditions can a quasinilpotent operator $T\in B(X)$ be determined to be nilpotent?
- 65
9
votes
1
answer
854
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 ...
- 585
4
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0
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146
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Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces
Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed?
The spaces in question include e.g.
\begin{equation}
X = (x: x \in l_2: p_i(x) ...
- 593
1
vote
2
answers
192
views
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 ...
- 61
5
votes
1
answer
188
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 $\...
- 12.6k
3
votes
3
answers
777
views
Radon-Nikodym property for space of signed measures
Given a measurable space, the vector space of signed measures is a Banach space. Does it have the Radon-Nikodym property? What if the space is of a special type, such as a nice topological space with ...
- 2,247
5
votes
1
answer
216
views
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\|_{...
- 4,115
2
votes
1
answer
136
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Eigenvectors of the dual of positive irreducible operators
This question was previously posted on MSE.
Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with ...
- 333
0
votes
0
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80
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Continuity of linear map on tensor product spaces with different norm properties
I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...
- 177
6
votes
1
answer
251
views
Finite representability of $\ell_p$ in subspaces of $L_p(0,1)$
Let $M$ be a closed subspace of $L_p(0,1)$, $1<p<\infty$, $p\neq 2$.
Suppose that M contains copies of $\ell_p^n$ uniformly.
Does $M$ contain a copy of $\ell_p$?
The result is true for $p=1$, ...
- 4,461
7
votes
1
answer
284
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 ...
- 1,361
2
votes
1
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369
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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 $L_p(0,1)\supset L_\infty (0,1)$...
- 617
0
votes
0
answers
210
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Examples of non $w^{*}$-closed complemented subspaces of a dual Banach space that are also dual spaces
Let $Y$ be a complemented, but not $w^{*}$-closed, subspace of a Banach space $X$. It is known that certain such $Y$ are not dual spaces.
Question: What are interesting examples of subspaces of the ...
- 7,067
0
votes
1
answer
145
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Renorming on a separable Banach space
Let us consider the sequence space $c_0$ with the equivalent norm
$$\Vert x \Vert^2 = \max_{i\ge1} \vert x^i \vert^2 + \sum_{i=2}^{\infty} 2^{-i+1} \vert x^i \vert^2 $$
for $x=(x^1,x^2,\ldots)\in c_0$....
- 85
9
votes
4
answers
4k
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Is the space of Radon measures a Polish space or at least separable?
Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
- 657
0
votes
0
answers
113
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The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
10
votes
3
answers
741
views
Is there a version of Fischer-Riesz theorem for Banach space?
$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...
- 406
2
votes
1
answer
103
views
Is $B \cap L^2\big( (0,T) \times (0,1)\big)$ closed in $L^2\big( (0,T) \times (0,1)\big)$?
I need help proving that the set $B \cap L^2\big( (0,T) \times (0,1)\big)$ is a closed subset of $L^2\big( (0,T) \times (0,1)\big)$, where $B$ is defined as:
$$B=\Big\{x \in L^{\infty}\big(0,T;L^1(0,1)...
- 55
0
votes
1
answer
141
views
Infimum of norms of elements in a hyperplane
In a Banach space X, given a norm one bounded linear functional $f$ and $c\in \mathbb{C}\backslash \{0\}$, define $H = \big\{ x\in X \,\vert\, f(x) = c\big\}$ and $\inf H$ = $\inf_{h\in H} \|h\|$.
Is ...
- 307
8
votes
1
answer
1k
views
Compactness of the unit ball of a Banach space for topologies finer than the weak* topology
Let $(\mathcal{X} , \|\cdot \|_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|_{\mathcal{X}'}$ the dual norm and define also the topological dual $\...
- 2,306
2
votes
1
answer
296
views
An abstract characterisation of weak* topologies
Is there a way of endowing the unit ball $B_X$ of a Banach space $X$ (we may assume that $X$ is an AL space, if that helps) with a topology $\tau$, so that $\tau=\sigma(Y^*,Y)$ (the weak* topology) if ...
- 153
3
votes
3
answers
319
views
Do these properties characterize Hilbert spaces?
Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ isometric with $l_2^2$ such that $x\in E$. Does this property characterize a (...
- 1,361
0
votes
1
answer
154
views
Finite dimensionality of a subspace
Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds:
$$ \...
- 4,115
2
votes
0
answers
78
views
Analogy between quasi-injective modules & extensible Banach spaces
Let $X$ be a module. $X$ is said to be quasi-injective if every homomorphism $h:A\to X$ from any submodule $A\subseteq X$ has an extension to an endomorphism $\tilde{h}:X\to X$.
A module $X$ is quasi-...
- 2,605
0
votes
0
answers
77
views
Property (H) in the dual norm
Consider the Hilbert space $l_2$ with an equivalent norm
$$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \},$$
where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\Vert ...
- 85
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
1
vote
0
answers
50
views
Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(E, |\...
- 657
1
vote
1
answer
113
views
Is $I-S$ in my attempt of Fredholm alternative injective?
Let $E$ be a Banach space. Let $\mathcal K(E)$ be the space of all compact (bounded linear) operators from $E$ to $E$. For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. Let ...
- 657
0
votes
1
answer
138
views
Smoothness of a Hilbert space under an equivalent norm
Let us take the Hilbert space $l_2$ with an equivalent norm
$\Vert x \Vert = \max \{2 \Vert x \Vert_1, \Vert x \Vert_2 \}$, where $\Vert x \Vert_1 =( \sum_{n=2}^\infty x_n^2 )^{\frac{1}{2}}$ and $\...
- 85
0
votes
0
answers
168
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\...
- 2,605
2
votes
1
answer
433
views
Density of $w^*$-support points
I am looking for a simple proof of the following theorem — wasn't able to come up with one myself. Should be a use of the Bishop–Phelps theorem, in some way:
Let $X$ be a Banach space, $D \subset X^*$ ...
- 165
0
votes
1
answer
205
views
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\...
- 145
0
votes
1
answer
156
views
Finding weak LUR property of $C[0,1]$ with an equivalent norm
On the space $X=C[0,1]$, define a norm $||| f |||^2=\Vert f \Vert_{\infty}^2 + \Vert f \Vert_2^2$, where $\Vert \cdot \Vert_\infty$ is the sup norm on $C[0,1]$ space and $\Vert \cdot \Vert_2$ is the $...
- 85
1
vote
0
answers
82
views
Injective envelopes of 1-extensible spaces
Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...
- 2,605
10
votes
2
answers
490
views
Surjective linear isometries on $\ell_\infty(\mathbb{N})$
In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
- 1,073
5
votes
1
answer
386
views
Contact points for John's ellipsoid
Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.
If $E$ and $K$ have exactly $2n$ contact points, say $(\...
- 1,361
13
votes
1
answer
401
views
Is there a reflexive Banach space whose ball is not the convex hull of its extreme points?
Let $X$ be a reflexive Banach space. Then the convex hull of the extreme points of the unit ball is weakly dense by the Krein-Milman theorem and Kakutani's theorem. My question is, if there is an ...
- 633
5
votes
4
answers
362
views
Dual norm of a subspace of $\ell_\infty^3$
We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
- 1,879
1
vote
2
answers
541
views
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^*\...
- 1,879