Questions tagged [banach-spaces]

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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ODE in Banach space

Have I understood this correctly: So originally we consider the following partial differential equation: $$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...
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Does property (V) imply the Grothendieck property for dual Banach spaces?

A Banach space $X$ has property (V) whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to ...
M.González's user avatar
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An explicit description for a certain type of infinite-dimensional homogeneous polynomials

This is a side question from Infinite-dimensional "algebraic varieties". Denote by $X_p$ ($1 \le p \le \infty$) the Banach spaces of complex sequences with finite $p$-norm and limit $0$. ...
Zerox's user avatar
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5 votes
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Radon-Nikodym property in Diestel & Uhl: a definition clarification

I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly ...
JWP_HTX's user avatar
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1 answer
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Spectrum invariant under (generalised) transpose as operator on trace class operators

For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
Frederik Ravn Klausen's user avatar
1 vote
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Notation for the space of eventually-zero sequences

An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
HardyHulley's user avatar
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1 answer
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Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
Emilio Ferrucci's user avatar
2 votes
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Convergence in Hausdorff distance of intersection of closed linear subspaces with a given closed convex set

I've run into the following problem when doing some work with non-commutative metric spaces, which seems like something people may have thought about before but I can't find anything on this problem ...
Sean's user avatar
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Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure

Let $(X, | \cdot |)$ be a Banach space. I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\...
ViktorStein's user avatar
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1 answer
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$c_{0}$ has no boundedly complete basis

Recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is called boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup_{n}\|\sum_{i=1}^{n}a_{i}x_{i}\|<\infty$, the series $\...
Dongyang Chen's user avatar
9 votes
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Weak compactness in $\mathcal{F}(X)$

Let $(X,0)$ be a pointed metric space and let $\mathcal{F}(X)$ be the natural predual of ${\rm Lip}_0(X)$, the space of Lipschitz functions on $X$ that map $0$ to $0$; here $\mathcal{F}(X)$ is really ...
Tomasz Kania's user avatar
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Uniformly convex Banach spaces

Theorem. If $X$ and $Y$ are uniformly convex Banach spaces, then for $1<p<\infty$ the space $$ X\oplus_pY=X\times Y, \qquad \Vert(x,y)\Vert:=(\Vert x\Vert_X^p+\Vert y\Vert_Y^p)^{1/p} $$ is ...
Piotr Hajlasz's user avatar
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Can a non-reflexive space embed into a reflexive space?

My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
Shridhar's user avatar
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quasi-Banach function spaces are subspace of $L_p$

It is well-known that any Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_1[0,1]$, and I can find a reference that any quasi-Banach rearrangement-invariant function ...
user92646's user avatar
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15 votes
2 answers
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Distinguishing topologically weak topologies of Banach spaces

Are the weak topologies of $\ell_1$ and $L_1$ homeomorphic? Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. ...
Tomasz Kania's user avatar
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3 votes
2 answers
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Unicellular compact operators

An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...
Markus's user avatar
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4 votes
2 answers
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Are sequences in $\ell^1(\mathbb N_0)$ converging uniformly on convex weakly compact subsets of $c_0(\mathbb N_0)$ norm convergent?

I think the question as expressed in the title should be clear. I do not know whether there is a known "characterization" of the weakly compact convex sets in $c_0(\mathbb N_0)$ but testing ...
TaQ's user avatar
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Extreme case of K-interpolation

Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space $X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as $$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
pipenauss's user avatar
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3 votes
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Asymptotic uniform convexity conditions for subsets of the $B_X$

The following question is relatively straightforward and almost looks like an exercise from a textbook but I have no idea how to handle it. The problem is related to spaces with asymptotically ...
Kevin Beanland's user avatar
2 votes
0 answers
186 views

Uniform limit of pointwise limits of continuous functions

Let $X$ be topological spaces, $Y$ a metric space and $(f_n)_{n\in\mathbb{N}}$ a sequence of functions, with $f_n:X\rightarrow Y$ pointwise limit of continuous functions for each $n\in\mathbb{N}$. ...
Lorenzo's user avatar
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Dual space of Carleman functions

Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which $$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
Janik's user avatar
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$ f,g\in \mathrm{VMO} $ but $ f\cdot g\notin \mathrm{VMO} $

We say a function $ f\in L^1_{\mathrm{loc}}(\mathbb{R}) $ is in $\mathrm{BMO}(\mathbb{R})$ if $$\|f\|_{\mathrm{BMO}}=\sup_{I}\frac{1}{|I|}\int\limits_I |f(y)-f_I|\, dy<\infty$$ for all intervals $I\...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
106 views

Definition of $1$-spreading basis and spreading model

I recall two definitions from Banach space theory Definition 1. Let $E$ be a Banach space, then a basis $(e_n)_{n\in\mathbb{N}}$ of $E$ is called $1$-spreading if $$\left\|\sum_{i=1}^k a_i e_{m_i}\...
Lorenzo's user avatar
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1 vote
1 answer
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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 ...
Matheus Manzatto's user avatar
5 votes
4 answers
353 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?
A beginner mathmatician's user avatar
2 votes
1 answer
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Banach-Mazur distance between Schatten-$p$ classes

Let $M_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M_n$ define $\|A\|_p:=(Tr(A^*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\...
A beginner mathmatician's user avatar
1 vote
0 answers
146 views

Infinite matrices from $\ell^p$ to $\ell^{p/(p-1)}$ that are compact operators

I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : \ell^{p} \rightarrow \ell^{q}$ defined as $...
jrranalyst's user avatar
5 votes
0 answers
267 views

Completeness of the space $L^p$ and the Axiom of Countable Choice

I am thinking about the proof that the usual $L^p$ spaces are complete. So, let $(X,\mathcal{F},\mu)$ be a measure space and let $p\in[1,+\infty)$. Important: by a measure I mean a nonnegative $\sigma$...
Ivan Feshchenko's user avatar
1 vote
1 answer
140 views

Charts for the Banach manifold of smooth almost complex structures $\mathcal{J}^{l}$

Consider the closure in the $C^l$-topology of the space of smooth almost complex structrues of a symplectic manifold $(M,\omega)$. We will denote this space by $\mathcal{J}^l$. It's a very used fact ...
Someone's user avatar
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1 vote
0 answers
188 views

Closure of finite rank operators on $L^p$

It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators. Besides this, the results by Per Enflo 1973 shows that this results is ...
Guy Fsone's user avatar
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4 votes
0 answers
205 views

"Cyclic vector" of sequence of operators

I recently encountered the following somewhat random-looking problem in my research. At first I thought that should not be too hard, but now, the more I think about it, the more interesting it seems. ...
Matthias Ludewig's user avatar
2 votes
0 answers
129 views

How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?

I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...
pyroscepter's user avatar
10 votes
1 answer
336 views

Group of isometries of Banach spaces a topological group?

Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$. Q: Is $\mathrm{Iso}(X)$ a topological group ...
Matthias Ludewig's user avatar
0 votes
1 answer
189 views

When does $C_b(X)$ admit a Schauder Basis?

Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that $$ \left\{d(x_n,\cdot)-d(x_0,\cdot)\...
Carlos_Petterson's user avatar
1 vote
1 answer
109 views

Bayesian inverse problems on non-separable Banach spaces

I am now studying Bayesian inverse problems. In the note of Dashti and Stuart https://arxiv.org/abs/1302.6989, they mentioned that "... when considering a non-separable Banach space $B$, it is ...
T. Huynh's user avatar
2 votes
0 answers
121 views

Fourier type of asymptotic-$\ell_{2}$ Banach spaces

A Banach space $X$ is said to have Fourier type $p\in[1,2]$ if the Fourier transform $\hat{f}(s):=\int_{\mathbb{R}}e^{-ist}f(t)dt$ defines a bounded linear operator from $L_{p}(\mathbb{R},X)$ to $L_{p'...
JWP_HTX's user avatar
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3 votes
1 answer
327 views

Bases in $c_{0}$

$c_{0}$, the space of the scalar sequence that converges to $0$ endowed with the sup norm, has two well-known bases: the unit vector basis $(e_{n})_{n}$, where $e_{n}(k)=1$ if $k=n$ and $0$ otherwise, ...
Dongyang Chen's user avatar
1 vote
1 answer
219 views

An approximation property in a separable topological vector space

Let $X$ be a topological vector space. Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
ABB's user avatar
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0 votes
0 answers
101 views

A question on the Haar basis for $L_{1}[0,1]$

Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. It is important to know the exact expression of the norm of $\|\sum_{i=1}^{n}a_{i}x_{i}\|$ for all $n$ and all scalars $a_{1},a_{2},\ldots,...
Dongyang Chen's user avatar
1 vote
1 answer
132 views

Isomorphy between Lebesgue space $L_1$ and the $l_1$ sum of $L_1[0,1]$ spaces

Is it true that for an infinite index set $I$, we have that $L_{1}([0,1]^{I}, \mathbb{R})$ can be written as the infinite direct sum of $L_{1}([0,1], \mathbb{R})$, i.e. $$L_{1}([0,1]^{I}, \mathbb{R})=\...
user44155's user avatar
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4 votes
0 answers
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$L_1$-subspace of the predual of a von Neumann algebra

If $M$ is a type $II$ von Neumann algebra, then the predual has a complemented subspace isometric to $L_1(0,1)$. It follows from the existence of expectation. However, I don't know whether such a ...
user92646's user avatar
  • 617
1 vote
1 answer
118 views

Quantifications of boundedly complete bases

Let $(x_{n})_{n=1}^\infty$ be a bounded sequence in a Banach space $X$. We set $$\textrm{ca}((x_{n})_{n=1}^\infty)=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$ Then $(x_{n})_{n=1}^\infty$ is norm-Cauchy ...
Dongyang Chen's user avatar
3 votes
1 answer
137 views

Quantifying shrinking bases

Let $X$ be a Banach space and let $(x_{n})_{n=1}^\infty$ be a (Schauder) basis for $X$. Let $(x^{*}_{n})_{n=1}^{\infty}$ be the biorthogonal functionals associated to the basis $(x_{n})_{n=1}^\infty$. ...
Dongyang Chen's user avatar
12 votes
1 answer
692 views

Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space?

Background: It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ ...
Orr Shalit's user avatar
4 votes
0 answers
177 views

Weak* HI Banach spaces

The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable. (See the previous posts Decomposable Banach Spaces, Hereditarily ...
S Argyros's user avatar
  • 771
2 votes
1 answer
134 views

On the symmetric basic sequence of a symmetric sequence space

Let $E$ be a separable Banach space with symmetric basis $\{e_i\}$ (it is also called a symmetric sequence space). Let $\{x_i\}$ be a normalized disjoint sequence in $E$, i.e., $\lVert x_i\rVert_E=1$ ...
user92646's user avatar
  • 617
1 vote
0 answers
94 views

Regularity of functions everywhere approximable by $n$-th degree polynomials

Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces. A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \}$...
Kacper Kurowski's user avatar
8 votes
0 answers
229 views

A question related to the separable quotient problem

I have the following question related to the previous posts Hereditarily indecomposable Banach spaces and Separable Quotient problem and Weak star separable and separable quotient problem Question....
S Argyros's user avatar
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2 votes
1 answer
105 views

Commutation of linear maps and extreme points

Let $X,Y$ be real vector spaces, $T: X\to Y$ be a linear map, and fix a nonempty $S\subseteq X$ (we do not assume that $S$ is neither convex nor compact (indeed, right now we do not assume any ...
Paolo Leonetti's user avatar
2 votes
2 answers
165 views

Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces

Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that: For $n \geq 0$, let $E_{n}$ ...
ScienceAge's user avatar

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