All Questions
Tagged with fa.functional-analysis banach-spaces
122 questions
12
votes
1
answer
1k
views
Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$
Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...
- 4,729
5
votes
1
answer
699
views
Can $L^1_{loc}$ be represented as colimit?
Let $L^1_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1_{m_K}$ denote the space ...
- 5,405
16
votes
1
answer
2k
views
What (classes of) Banach spaces are known to have Schauder basis?
Motivation:
I am trying to see for what class of Banach spaces the following result is true:
There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
- 365
12
votes
2
answers
3k
views
Direct proof of injectivity of $L_\infty$
I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
$L_\infty$ as ...
- 1,697
10
votes
3
answers
739
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
81
votes
3
answers
9k
views
Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 31.5k
43
votes
1
answer
5k
views
Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
- 559
34
votes
1
answer
3k
views
tr(ab)=tr(ba), part 2.
This is a Banach space version of Andre Henriques' question
Trace Question
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
- 31.5k
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 7,902
23
votes
9
answers
2k
views
Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
21
votes
2
answers
1k
views
Meager subspaces of a Banach space and weak-* convergence
I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to ...
- 29.7k
19
votes
1
answer
3k
views
Infinite convex combinations in a Banach space
Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers $(\...
- 60.5k
16
votes
2
answers
682
views
Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?
Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, a\...
- 3,497
14
votes
2
answers
6k
views
Are weak and strong convergence of sequences not equivalent?
For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim_{i\to\...
- 3,584
12
votes
3
answers
1k
views
Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
- 4,461
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 4,878
11
votes
4
answers
1k
views
Example of noncomplete quotient of complete lcs mod closed subspace
The following statement is well-known: for a Fréchet space $V$ and a closed subspace $W \subseteq V$ the quotient $V / W$ is again complete and hence a Fréchet space. For the particular case of a ...
- 8,098
8
votes
0
answers
1k
views
Strictly singular operators and their adjoints
This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...
- 1,073
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
7
votes
1
answer
246
views
A notion of restricted injectivity for Banach spaces
I apologize in advance if this is well-known.
Let $X$ be a Banach space. Let's call only for this post that $X$ is self-injective if for every closed subspaces
\begin{equation}
A\subseteq B\subseteq X ...
- 2,605
5
votes
0
answers
228
views
What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
- 41.8k
4
votes
2
answers
558
views
Is a specific sequentially closed subset of $M([0,1])$ closed?
Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
- 95
3
votes
1
answer
561
views
Construction of an infinitely Fréchet differentiable function with given set of zeros in a Banach space
After looking at this question, I am now wondering if the following is true.
Let $X$ be a separable Banach space over $\mathbb R$ or $\mathbb C$, and $A\subseteq X$ a closed set. Then there exists ...
- 1,755
30
votes
3
answers
3k
views
Surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
- 301
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
21
votes
5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 629
21
votes
3
answers
3k
views
Can you tell whether a space is Banach from the unit ball?
Let $V$ be a real vector space. It is well known that a subset $B\subset V$ is the unit ball for some norm on $V$ if and only if $B$ satisfies the following conditions:
$B$ is convex, i.e. if $v,w\...
- 8,493
19
votes
3
answers
1k
views
What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?
A colleague asked me the following question:
"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"
This ...
- 356
18
votes
1
answer
564
views
Is the space of Hankel operators complemented in B(H)?
Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$.
Let ...
- 25.8k
18
votes
3
answers
2k
views
What are the right categories of finite-dimensional Banach spaces?
This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
- 11.4k
17
votes
1
answer
912
views
$(1+\epsilon)$-injective Banach spaces, complex scalars
It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
- 1,704
16
votes
6
answers
2k
views
Finding closed subspaces whose sum isn't closed
Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum ...
- 42.8k
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 161
15
votes
1
answer
4k
views
Is there a simple direct proof of the Open Mapping Theorem from the Uniform Boundedness Theorem?
The Open Mapping Theorem, the Bounded Inverse Theorem, and the Closed Graph Theorem are equivalent theorems in that any can be easily obtained from any other. The Closed Graph Theorem also easily ...
- 2,049
15
votes
1
answer
889
views
Operator norms of circulant matrices
The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix.
For complex numbers $a_1,\ldots,a_n$, I will use the notation
$$
\mbox{...
- 717
15
votes
3
answers
8k
views
What is an isomorphism of Banach spaces?
The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
- 26.8k
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
14
votes
4
answers
550
views
About the existence of characters on $B(X)$
Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M_n(\mathbb{C})$ ...
- 355
13
votes
1
answer
724
views
Trace-class operator satisfies $\sum |\lambda_n|<\infty$?
Here's an "exercise" which I thought should be easy, but which I find myself unable to do.
Let $V$ be a Banach space.
Recall that an operator $f:V\to V$ is trace-class if it is in the image of the ...
- 43.2k
13
votes
1
answer
911
views
Are $L^\infty(\Bbb R)$ and $L^2(\Bbb R)$ homeomorphic?
It's easy to see that, for $1\le p,q< \infty$ the spaces $L^p(\Bbb R)$ and $L^q(\Bbb R)$ of $p$-th and $q$-th power integrable functions on the real line are homeomorphic as topological spaces. In ...
- 3,017
12
votes
1
answer
908
views
Equivalence of σ-convex hull and closed convex hull
Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as
$$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\,...
- 123
11
votes
5
answers
5k
views
A criterion for the sum of two closed sets to be closed ?
Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ ...
- 452
11
votes
1
answer
964
views
Quotients of l^infty
Let $M$ be a closed subspace of $l^\infty$. Suppose that the quotient $l^{\infty}/M$ is isomorphic to $l^\infty$. Is it true that $M$ is complemented in $l^\infty$?
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
11
votes
3
answers
1k
views
Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 180
11
votes
3
answers
445
views
Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?
Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...
- 662
10
votes
2
answers
666
views
Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
10
votes
0
answers
226
views
Extremal bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
- 4,535
10
votes
1
answer
899
views
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
- 710
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k