All Questions
103 questions
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
4
votes
0
answers
508
views
Good reference for noncommutative $L^p$ spaces
I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...
- 41
4
votes
0
answers
90
views
$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology
This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
3
votes
1
answer
269
views
What is a standard name for this kind of unconditional bases in Banach spaces?
I am looking for a standard name (if it exists) for the following property of a Schauder basis $(e_i)_{i=1}^\infty$ in a Banach space $X$:
$$\|\sum_{i\in F}x_ie_i\|\le\|x\|$$for any $x=\sum_{i=1}^\...
- 41.9k
3
votes
3
answers
489
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
$$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
- 799
3
votes
1
answer
260
views
Reference request: completion of Banach norm on sum
Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset
$$
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\},
$$
a is ...
- 5,405
3
votes
1
answer
221
views
Does Bishop-Phelps Theorem hold for extreme points (slightly different version)?
Recall the Bishop-Phelps Theorem.
Bishop-Phelps Theorem: Let $B\subseteq E$ be a bounded, closed, convex subset of a real Banach space $E.$
Then the set
$$\{e^*\in E^*: e^* \text{ attains its ...
- 623
3
votes
1
answer
280
views
example of an $\ell_1$-saturated Banach space without an unconditional basis
Giorgos Petsoulas, in his paper "A class of $\ell^p$ saturated Banach spaces," has constructed for each $1<p<\infty$ a space $\mathfrak{X}_p$ which is complementably $\ell_p$-saturated but ...
- 1,591
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
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
3
votes
1
answer
245
views
Orthonormal basis in $\ell^n_p$
Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of
$$
\sum_{i=1}^k \|a_i\|_{\ell^p}^2
$$
for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis.
...
- 4,432
3
votes
2
answers
435
views
A possible norm on a subspace of $C^\infty([0,1])$?
I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...
- 3,388
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
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
- 183
3
votes
1
answer
502
views
Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
- 11.5k
3
votes
0
answers
282
views
Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$
Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...
- 2,605
2
votes
2
answers
602
views
Reference for weak*-semigroup
Let $X$ a dual Banach space (there exists a Banach space $Y$ such that $X=Y'$).
A weak* semigroup on $X$ is a semigroup $(T_t)_{\geq 0}$ on $X$ such that, for all $x\in X$, we have $T_tx\to x$ in the ...
- 1,222
2
votes
1
answer
104
views
Operators "building" linear independant sets
Let $E$ be a separable Banach space and let $T\in L(E,E)$.
Is there a condition on $T$ ensuring that:
$$
\mbox{$\{x_n\}_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow
\{T(x_n)\}_{n=1}^N\cup \...
- 93
2
votes
1
answer
240
views
BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$
Let $\ell_p^n$ be the $n$-dimensional real or complex $\ell_p$-space and let $\mathscr{B}(\ell_p^n)$ be the space of matrices on $\ell_p^n$ endowed with the operator norm. I am looking for any ...
- 11.3k
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
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
2
votes
2
answers
406
views
"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 ...
- 1,461
2
votes
1
answer
132
views
Form of finite dimensional contractive projection in $L_p$
Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form:
$Pf = \sum_{k=1}^n g_k \int h_kf$
Where $\|g_k\|_p = \|h_k\|_q = \...
- 311
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
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
0
answers
150
views
Non-separable asymptotic $\ell_1$ space
The Figiel-Johnson Tsirelson space is an example of an asymptotic $\ell_1$ Banach space not containing $\ell_1$. The notion of asymptotic $\ell_1$ is with respect to some basis, but a coordinate free ...
user114263
2
votes
0
answers
93
views
Open problems concerning Araujo's biseparating maps
Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$
Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
- 623
2
votes
0
answers
189
views
Dunford−Pettis property of $L^1(\mu)$
$\def\bs#1{\boldsymbol#1}\def\sp{\kern.4mm}$Let $\bs K$ be either the standard real or complex topological field, and let $E$ be a Hausdorff locally convex space over $\bs K\sp$. Then saying that $E$ ...
- 3,584
2
votes
0
answers
141
views
Quotients in complex interpolation of Banach spaces
Let $(X_0,X_1)$ be an admissible pair of complex Banach spaces with $X_0$ continuously embedded in $X_1$. For $0<\theta<1$, let us denote by $X_\theta =(X_0,X_1)_\theta$ the complex ...
- 4,461
2
votes
0
answers
106
views
Type-cotype inequalities for arbitrary orthonormal systems
Let $X$ be a B-convex Banach space and let $v^1 = (v^1_1,…,v^1_n), …, v^n = (v^n_1,…,v^n_n)$ be an orthonormal basis of $\mathbb{R}^n$. My question is what one can say about $\left( \sum_i \Vert \...
- 1,163
2
votes
0
answers
346
views
When is the sum of complemented subspaces complemented?
Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
- 935
2
votes
0
answers
117
views
Maximum Principle with Banach Control Space
This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...
- 197
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
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
1
vote
1
answer
201
views
Reference request for sums of Grothendieck spaces
I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell_p$...
- 125
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
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
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
$$ \Vert (x_n) \Vert_{p,w} := \sup_{\...
- 25.8k
1
vote
1
answer
221
views
Kernel of a convex combination of projections
Assume that we have finitely many projections $P_1,\dots, P_n$ on a Banach space $X$ (take an explicit case of $X=L_p(\mu)$ if a concrete examples is better). Consider their convex combination $P=\...
- 11
1
vote
2
answers
181
views
Where can I find some articles and lecture notes in renorming theory in Banach spaces? [closed]
I am really into renorming theory in Banach spaces especially, renorming in non-reflexive Banach spaces such that they have nice property, for example they have fixed point property,locally uniformly ...
- 823
1
vote
1
answer
184
views
Special kind of operators
Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$
where $(...
- 9,853
1
vote
0
answers
82
views
Injective envelopes of 1-extensible spaces
Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...
- 2,605
1
vote
0
answers
47
views
Existence for a nonlinear evolution equation with a monotone operator that is not maximal
We consider the nonlinear evolution equation
$$
\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
1
vote
0
answers
111
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
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
1
vote
0
answers
174
views
Estimates of entropy of functional spaces
Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it.
...
- 21.8k
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
0
votes
1
answer
178
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
0
votes
1
answer
233
views
Is $(\ell^1(\mathbb N_0),\sigma(\ell^1,\ell^\infty))$ not quasi-complete?
In Jarchow's Locally Convex Spaces this not being quasi-complete is asserted on page 206 referring to Corollary 11.4.4 on page 228 saying that a Banach space is reflexive if and only if its closed ...
- 3,584
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 ...
- 7,067
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