All Questions
Tagged with fa.functional-analysis banach-spaces
1,222 questions
5
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1
answer
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Weak convergence implying norm convergence
A surprising (to me) consequence of Hahn-Banach is that when a sequence converges weakly then there is another sequence made of (finite) convex combination which converges in norm (to the same element)...
- 31.5k
1
vote
0
answers
149
views
Banach spaces with simple best approximate solutions
Let $\langle V,||.||\rangle$ be a Banach space such that:
$\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$
$\;\;$ that ...
CommunityBot
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15
votes
1
answer
1k
views
Intersection of complemented subspaces of a Banach space
The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here.
Question. Let $X$ be a Banach ...
- 60.5k
4
votes
1
answer
287
views
Second conjugate operators to operators on $c_0$
I posted my question at MS but unfortunately it is still without a response, so let me ask it here.
We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,...
- 31.5k
7
votes
1
answer
682
views
$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$
Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...
- 31.5k
17
votes
4
answers
2k
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Banach-Mazur applied to a Hilbert space
The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm.
...
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3
votes
2
answers
461
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Complemented subspaces isomorphic to $c_0$ in $\mathcal{B}(E)$ [closed]
It is well known that neither
1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$
nor
2) $c_0$ is a quotient of $\mathcal{B}(H)$
for a Hilbert space $H$. Can we replace $H$ above ...
CommunityBot
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9
votes
1
answer
893
views
Perturbations of an operator that disconnect the spectrum
The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under ...
CommunityBot
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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
15
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3
answers
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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 ...
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0
votes
1
answer
498
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Quotient of \ell_1 by space of finite sequences
The following question came up during a reading of Rudin's functional analysis. I have not been able to find any information through searching online, but I apologise if the answer is obvious, or the ...
CommunityBot
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34
votes
1
answer
3k
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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
0
votes
2
answers
225
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Codimension of $J(\omega_1)$ in its bidual
I am reading the paper
G. A. Edgar, A long James space, in: Measure Theory, Oberwolfach 1979, Lectures Notes in Math. 794, Springer-Verlag (1980) pp. 31-37.
and I am pretty confused by the remarks ...
- 41.1k
1
vote
1
answer
293
views
Basic sequences
Nowadays, we know that there exist Banach spaces without unconditional basic sequences. Do we know if something a bit milder holds? Namely, is that true each non-reflexive Banach space contains a ...
- 31.5k
12
votes
2
answers
547
views
Balls in spaces of operators
I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try.
Is ...
- 2,378
6
votes
2
answers
2k
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How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
- 2,135
6
votes
0
answers
3k
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Projective and injective tensor product
It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space
$(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$.
If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
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2
votes
1
answer
323
views
Recovering Schauder decompositions
The problem of Schauder decomposition of a given Banach space seems to play an important role in the geometry of Banach spaces, especially when one is interested in finite dimensional Schauder ...
- 31.5k
4
votes
2
answers
484
views
When is a metric space isometrically embeddable into some Banach space?
EDIT
Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something ...
- 28.6k
1
vote
1
answer
312
views
Invertibility of frame/sampling operator on Bargmann-Fock spaces
Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
16
votes
2
answers
4k
views
Usefulness of Frechet versus Gateaux differentiability or something in between.
If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux ...
- 399
4
votes
3
answers
3k
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Examples of Banach spaces and their duals
There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued ...
- 159
7
votes
3
answers
814
views
Preduals of B(E)
For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always a ...
- 18.7k
1
vote
0
answers
369
views
Infinite internal direct sums of subspaces
Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a ...
- 11.3k
7
votes
1
answer
423
views
Best constant in comparison between Rademacher and gaussian averages?
Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables.
What is the best ...
- 31.5k
0
votes
1
answer
365
views
Integral in a σ−convex set.
Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
CommunityBot
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7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
CommunityBot
- 1
6
votes
1
answer
404
views
Unique preduals up to (nonisometric) isomorphism?
It's well known that there are Banach spaces which has a unique isometric predual-- for example, any von Neumann algebra. As other questions on here (for example, Isomorphisms of Banach Spaces ) ...
CommunityBot
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6
votes
1
answer
1k
views
Must a surjective isometry on a dual space have a pre-adjoint?
Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^*$ is a surjective isometry of $X^*$.
In general, a linear map $g:X^* \...
- 7,329
0
votes
0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
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22
votes
3
answers
7k
views
Subspace of $L^2$ that lies in $L^\infty$
Let $E$ be a closed subspace of $L^2[0,1]$. Suppose that $E\subset{}L^\infty[0,1]$. Is it true that $E$ is finite dimensional?
PS. This is actually a question from the real analysis qualifier. I came ...
- 25.8k
3
votes
1
answer
1k
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Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
CommunityBot
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4
votes
1
answer
2k
views
Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...
- 744
7
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1
answer
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If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
CommunityBot
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1
vote
4
answers
614
views
Variants of point fixed theorem
Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$.
...
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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 ...
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2
votes
2
answers
1k
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Are coordinate functions on topological vector spaces always continuous?
Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{...
- 31.5k
7
votes
3
answers
1k
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Non-Borel subspace of Banach space
Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
CommunityBot
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4
votes
1
answer
221
views
existence of charaterization of amenable groups by complementation?
Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...
- 18.7k
6
votes
1
answer
453
views
The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
- 11.4k
1
vote
1
answer
506
views
Bessel sequence, uniformly minimal, separated
Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded ...
- 31.5k
4
votes
2
answers
676
views
Basis for L_infty(R)
Let $V$ be the Banach space of bounded sequences of reals with the sup norm. Does there exists a subset $B$ of $V$ such that
Linear Independence: For all functions $c$ in $\mathbb{R}^B$, if $\sum_{b ...
- 48.8k
4
votes
2
answers
707
views
Selecting basic sequences
Suppose $(x_\alpha)_\alpha$ is an uncountable, linearly independent family of norm one vectors in a Banach space. Can one always select a basic sequence (or at least a minimal system) from this family?...
- 355
3
votes
1
answer
615
views
When is a fixed point of f^n a fixed point of f?
Let $E$ be a Banach space and $f:E\to E$ be a continuous map. By $f^n$ we denote the $n$-th iterate of $f$, i.e. $f^n:=\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}}$. Let $x_0$ denote a ...
- 12.6k
8
votes
1
answer
713
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Factoring operators $L_\infty \longrightarrow L_2$ as the composition of $n$ strictly singular operators, $n\in \mathbb{N}$
Motivation and background This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $\mathcal{B}(L_\infty)$ of all (bounded, linear) operators on $L_\...
- 31.5k
7
votes
1
answer
1k
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Banach spaces with a certain separability property
In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset $...
- 1,544
6
votes
0
answers
2k
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Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
11
votes
1
answer
654
views
Nonseparable Hilbert spaces as quotients of spaces of bounded functions
Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember ...
- 31.5k
7
votes
2
answers
808
views
Is a subspace with a certain property dense in the dual of a vector space?
Suppose we have a normed vector space $V$ and its dual $V^*$, and suppose that $X \subseteq V^*$ has the property that for every $v \in V$, there is some $\phi \in X$ with $\Vert \phi \Vert = 1$ such ...
- 4,060
2
votes
2
answers
354
views
A bound on linear functionals over cotype 2 spaces
This is a modification of the somewhat naive question that I asked below.
Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
- 31.5k