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7 votes
1 answer
415 views

Is there a “Closure-of-Range Theorem” for Banach spaces?

The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions: $T(X)$ is $s$-closed; $T(X)$ is $...
Pietro Majer's user avatar
  • 60.5k
2 votes
0 answers
82 views

What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$

Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping $$ f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X, $$ with $f(0)=0$ is a radial and maps ...
shuhalo's user avatar
  • 5,327
1 vote
1 answer
209 views

Rate of convergence of mollified functions in $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
Akira's user avatar
  • 835
0 votes
1 answer
106 views

Convergence of mollified functions in weighted $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
Akira's user avatar
  • 835
2 votes
1 answer
148 views

Is projection of a closed subspace Borel?

Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
TaQ's user avatar
  • 3,584
6 votes
1 answer
290 views

Analytic maps on Banach spaces: analyticity upgrade

Consider the following problem. Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and $$ f:U\to G $$ an analytic map, such ...
Lorenzo Pompili's user avatar
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 ...
Onur Oktay's user avatar
  • 2,605
1 vote
0 answers
47 views

Existence for a nonlinear evolution equation with a monotone operator that is not maximal

We consider the nonlinear evolution equation $$ \dot{u}(t) + Bu(t) = 0, \quad u(0)=0 $$ with $$ A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
ChocolateRain's user avatar
0 votes
0 answers
208 views

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 ...
Jon Bannon's user avatar
  • 7,057
8 votes
1 answer
641 views

Reference Request: Arzelà-Ascoli for Hölder norm

I'm studying the Banach Space of Hölder continuous functions $f:[0,1]\to\mathbb{R}^{+}$ with a parameter $\alpha$. In this space, I consider the usual Hölder norm $\|\cdot\|_\alpha$ and I'm looking ...
NewGuy23's user avatar
1 vote
1 answer
262 views

Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable ...
Analyst's user avatar
  • 657
3 votes
1 answer
181 views

General form of bounded linear functionals on Banach spaces

It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$. On p.188 of Introductory ...
user90041's user avatar
  • 709
1 vote
0 answers
110 views

Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space

$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set. For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
Overflowian's user avatar
  • 2,533
4 votes
1 answer
281 views

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
  • 4,461
1 vote
1 answer
452 views

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
4 votes
2 answers
428 views

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
  • 3,584
3 votes
0 answers
282 views

Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$

Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations. Let $w_0(A)$ be the subspace of all weakly null sequences in $...
Onur Oktay's user avatar
  • 2,605
-1 votes
1 answer
120 views

Definition of a $\psi$-Banach space [closed]

Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
Motaka's user avatar
  • 291
5 votes
1 answer
244 views

Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$?

$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ ...
Miek Messerschmidt's user avatar
10 votes
2 answers
489 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 $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
Kevin Beanland's user avatar
2 votes
0 answers
85 views

Functions with smooth projections on finite-dimensional subspaces

Let $E,F$ be Banach spaces and $F$ be finite-dimensional and $E$ be strictly convex. Let $f\in C(F,E)$ have the property that: $$ \text{For every finite-dimensional subspace $E'\subseteq E$ we have } ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
104 views

Operators "building" linear independant sets

Let $E$ be a separable Banach space and let $T\in L(E,E)$. Is there a condition on $T$ ensuring that: $$ \mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow \{T(x_n)\}_{n=1}^N\cup \...
TomCat's user avatar
  • 93
4 votes
1 answer
279 views

Reference request: Baire's theorem for operator ranges

Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
Jochen Glueck's user avatar
3 votes
1 answer
178 views

Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists

The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as $$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
Gilles Bonnet's user avatar
3 votes
1 answer
951 views

Specific criterion for the sum of two closed sets to be closed

Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$. I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\...
Westlife's user avatar
4 votes
0 answers
146 views

When does an operator from $\ell_1$ to itself factor through $\ell_p$?

I would like to know whether a given operator from $\ell_1$ to itself, given by a matrix $A$, factors through $\ell_p$, for $p>1$.Does anyone know any references/results on this topic? I am ...
Gamabunto's user avatar
6 votes
3 answers
852 views

Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram $\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 ...
santker heboln's user avatar
0 votes
0 answers
154 views

Use of this space of very rapidly decreasing continuous functions

Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space $$ V_p := \left\{ f \in C([0,\infty)):\, \sum_{n=1}^{\infty} ...
ABIM's user avatar
  • 5,405
0 votes
1 answer
177 views

Convergence in LB-spaces

Edit: Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
ABIM's user avatar
  • 5,405
3 votes
1 answer
260 views

Reference request: completion of Banach norm on sum

Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset $$ X_1+X_2 = \left\{ x\in X:\, x= \beta_1 x_1 + \beta_2 x_2 \, \beta_i \in \mathbb{R},\, x_i \in X_i \right\}, $$ a is ...
ABIM's user avatar
  • 5,405
5 votes
1 answer
224 views

reference request: unbounded operators on normed spaces

I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' functional ...
F. Carbon's user avatar
  • 105
4 votes
1 answer
352 views

Minimality properties of James' space

I am interested in the following question about James' quasi-reflexive Banach space $\mathcal{J}$: Does there exists a non-Hilbertian subspace $X$ of $\mathcal{J}$ such that $X$ isomorphically ...
N. de Rancourt's user avatar
6 votes
0 answers
99 views

Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?

The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature. Can every ...
Tomasz Kania's user avatar
  • 11.3k
1 vote
1 answer
120 views

Does taking the modulus preserve weak $p$-summability of sequences in $L_q$?

For this question, all Banach spaces are over the reals. Let $1\leq p<\infty$. Recall that a sequence $(x_n)$ in a Banach space $E$ is weakly $p$-summable if $$ \Vert (x_n) \Vert_{p,w} := \sup_{\...
Yemon Choi's user avatar
  • 25.8k
9 votes
4 answers
4k views

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 ...
Mark's user avatar
  • 657
4 votes
1 answer
189 views

Intrinsic volumes of non-polyconvex, non-compact sets

I am reposting this question I asked and bountied on Math SE, which has been upvoted but not answered or commented on. The intrinsic volumes (AKA Minkowski Functionals or, with different ...
Joe Previdi's user avatar
10 votes
1 answer
203 views

Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski

This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...
Yemon Choi's user avatar
  • 25.8k
6 votes
2 answers
240 views

Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE. Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
erz's user avatar
  • 5,529
11 votes
1 answer
336 views

Notions in the literature capturing the "symmetric" or "homogeneous" flavour of $L_p$?

This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise. For $E$ a Banach space, $K(E)$ and $B(E)$ will denote the Banach algebras of ...
Yemon Choi's user avatar
  • 25.8k
5 votes
1 answer
158 views

What is a name for co-Sobczyk Banach spaces?

Definition. Let us define a Banach space $X$ to be co-Sobczyk if every linear bounded operator $T:Z\to c_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c_0$. ...
Taras Banakh's user avatar
  • 41.8k
4 votes
1 answer
136 views

Defining a topology by sequences

Suppose we have a Banach space $X$ and have chosen a set $\Sigma$ consisting of some sequences whose members are in $X$. We can then say that $(x_n)_{n=1}^\infty\in X^\mathbb{N}$ is $\Sigma$-...
user avatar
2 votes
1 answer
245 views

"Compactness in measure" in function spaces

In Chapter 4.9 of the book "Measures of noncompactness and condensing operators" (Vol. 55 of *Operator theory: advances and applications), the authors mention the property "compactness ...
Matt R.'s user avatar
  • 163
3 votes
3 answers
488 views

Sum of subspaces is closed iff inclination is positive

It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination $$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
Christian's user avatar
  • 799
1 vote
1 answer
176 views

Reference on vector-valued convex conjugate

The following definition of convex conjugate is taken from Wiki: Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot ,...
Idonknow's user avatar
  • 623
6 votes
1 answer
270 views

Approximation property counterexamples? (Also: relation to tensor products)

I remember reading somewhere (but unfortunately, I've forgotten where it was) that the canonical map from the (completed) projective tensor product of two Banach spaces to the (completed) injective ...
Jeff Egger's user avatar
3 votes
1 answer
221 views

Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?

Recall the Bishop-Phelps Theorem. Bishop-Phelps Theorem: Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$ Then the set $$\{e^*\in E^*: e^* \text{ attains its ...
Idonknow's user avatar
  • 623
8 votes
1 answer
232 views

Lipschitz right inverses of Banach space quotients

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear). Of course, if $Y$ is ...
Miek Messerschmidt's user avatar
1 vote
0 answers
217 views

Status of an open problem in isometric aspect of Banach space theory

The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$ Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach ...
Idonknow's user avatar
  • 623
7 votes
0 answers
327 views

Status of two Banach space theory open problems posted by Pełczyński

In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems. Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is ...
Idonknow's user avatar
  • 623
5 votes
1 answer
1k views

Reference request: The resolvent is analytic in the resolvent set

I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators. On page 192, he defines the resolvent and spectrum of $T$: Later on in the paragraph, he then proceeds by ...
user860374's user avatar