All Questions
103 questions
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 $...
- 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 ...
- 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 ...
- 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 ...
- 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 $E\,$? ...
- 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 ...
- 1,075
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
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_{\{||\...
- 171
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 ...
- 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 ...
- 81
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 ...
- 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 ...
- 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 ...
- 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 ...
- 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}
$$
is ...
- 28k
4
votes
2
answers
428
views
Are sequences in $\ell^1(\mathbb N_0)$ converging uniformly on convex weakly compact subsets of $c_0(\mathbb N_0)$ norm convergent?
I think the question as expressed in the title should be clear. I do not know whether there is a known "characterization" of the weakly compact convex sets in $c_0(\mathbb N_0)$ but testing ...
- 3,584
3
votes
0
answers
282
views
Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$
Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...
- 2,605
-1
votes
1
answer
120
views
Definition of a $\psi$-Banach space [closed]
Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...
- 291
5
votes
1
answer
244
views
Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$?
$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ ...
10
votes
2
answers
489
views
Surjective linear isometries on $\ell_\infty(\mathbb{N})$
In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
- 1,073
2
votes
0
answers
85
views
Functions with smooth projections on finite-dimensional subspaces
Let $E,F$ be Banach spaces and $F$ be finite-dimensional and $E$ be strictly convex. Let $f\in C(F,E)$ have the property that:
$$
\text{For every finite-dimensional subspace $E'\subseteq E$ we have } ...
- 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:
$$
\mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow
\{T(x_n)\}_{n=1}^N\cup \...
- 93
4
votes
1
answer
279
views
Reference request: Baire's theorem for operator ranges
Let $F$ be a Banach space. A vector subspace $U \subseteq F$ is called an operator range if there exists a Banach space $E$ and a bounded linear mapping $T: E \to F$ such that $TE=U$. By a quotient ...
- 12.5k
3
votes
1
answer
178
views
Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists
The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
- 183
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\...
- 33
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 ...
- 53
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 ...
- 1,338
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
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
- 5,405
0
votes
1
answer
177
views
Convergence in LB-spaces
Edit:
Let $X$ be a strict LB-space described by $\lim X_n$ and suppose that $\{x_n\}_{n \in \mathbb{N}}$ converges in $X$. I'm looking for a reference showing that $x_n$ must converge in some $X_N$.
- 5,405
3
votes
1
answer
260
views
Reference request: completion of Banach norm on sum
Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset
$$
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\},
$$
a is ...
- 5,405
5
votes
1
answer
224
views
reference request: unbounded operators on normed spaces
I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' functional ...
- 105
4
votes
1
answer
352
views
Minimality properties of James' space
I am interested in the following question about James' quasi-reflexive Banach space $\mathcal{J}$:
Does there exists a non-Hilbertian subspace $X$ of $\mathcal{J}$ such that $X$ isomorphically ...
- 1,005
6
votes
0
answers
99
views
Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?
The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.
Can every ...
- 11.3k
1
vote
1
answer
120
views
Does taking the modulus preserve weak $p$-summability of sequences in $L_q$?
For this question, all Banach spaces are over the reals.
Let $1\leq p<\infty$. Recall that a sequence $(x_n)$ in a Banach space $E$ is weakly $p$-summable if
$$ \Vert (x_n) \Vert_{p,w} := \sup_{\...
- 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 ...
- 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 ...
- 509
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 ...
- 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 ...
- 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 ...
- 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 $\bar T:X\to c_0$.
...
- 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 $\Sigma$-...
user78375
2
votes
1
answer
246
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 ...
- 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
$$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
- 799
1
vote
1
answer
176
views
Reference on vector-valued convex conjugate
The following definition of convex conjugate is taken from Wiki:
Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$
Denote the dual pairing by
$$\langle \cdot ,...
- 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 ...
- 508
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 ...
- 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 ...
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 ...
- 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 ...
- 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 ...
- 289