All Questions
Tagged with fa.functional-analysis banach-spaces
1,222 questions
4
votes
1
answer
819
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The Notion of Strong Measurability for Separable Banach Spaces
Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-...
- 23.2k
16
votes
3
answers
1k
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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 ...
- 423
1
vote
0
answers
439
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Does the dual Banach space $B(\ell^\infty)$ have weak* normal structure?
Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if
$$
\sup_{y\in K} \|x-y\|={\rm diam}(K).
$$
where ${\rm diam}(K)$ denotes the ...
- 11.3k
4
votes
2
answers
463
views
When $L^\infty$ is 1-injective
It is known that when $\mu$ is $\sigma$-finte measure, then $L^\infty(\mu)$ is $1$-injective.
But I want to know whether it is right for any $L^\infty$ spaces.
- 11.3k
2
votes
1
answer
344
views
Chebyshev centres of a bounded closed convex set in a strictly convex Banach space
Suppose $X$ is a strictly convex Banach space. Does there exist a bounded closed convex set $K$ in $X$ such that the set of all Chebyshev centers $C(K)$ of $K$ is a proper subset of $K$ with diameter ...
- 4,878
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 ...
CommunityBot
- 1
4
votes
2
answers
404
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Is the ideal of functions vanishing at a set complementable in $C(X)$?
Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$
$$
I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\}
$$
complementable (as a closed ...
- 31.5k
7
votes
0
answers
559
views
The Banach space of bounded functions with countable support
Let $X$ be a set of cardinality $\aleph_1$ and consider the Banach space $\ell_\infty^c(X)$ of all scalar-valued bounded functions on $X$ which are non-zero only for countably many elements of $X$ ...
- 11.3k
2
votes
1
answer
969
views
Positive definite quadratic forms on Banach spaces
This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if $E(x,x)\...
- 1,731
1
vote
0
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198
views
Measurability of a map that takes a functional to its composition with a linear operator
Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated.
Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with the ...
CommunityBot
- 1
1
vote
0
answers
281
views
Clarkson's inequalities for Banach space valued functions
In standard analysis, Clarkson's inequalities expresses the norms of the sum and difference of two functions in $L^p$ in terms of the norms of the individual functions. In particular, one may use the ...
- 18.7k
2
votes
2
answers
392
views
relation between of uniformly rotund in every direction and uniformly rotund and locally uniformly rotund
The norm of a Banach space $X$ is said to be uniformly rotund in every direction if $$\lim_{n→∞}\|x_n−y_n\|=0$$ whenever $$x_n,y_n∈SX$$ are such that $$\lim_{n→∞}\|x_n+y_n\|=2$$ and there is a $z∈X$ ...
2
votes
1
answer
188
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Existence of normal structure in strictly convex Banach spaces
Does there exists a strictly convex Banach space which is not uniformly convex and has normal structure ?
- 11.3k
6
votes
0
answers
307
views
a question about Tsirelson's space
NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.
...
- 1,591
1
vote
0
answers
170
views
A question about Smulian lemma
Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then
(i) $||.||$ is Frechet diffrentiable at $x$ iff $\lim\limits_{n\to\infty}||f_n-...
- 823
4
votes
1
answer
391
views
Frechet differentiable implies reflexive?
Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?
Can any one help me?
thanks
- 4,713
5
votes
0
answers
2k
views
Denseness of finite rank operators in $\mathcal{B}(X,Y)$
Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
https://math.stackexchange.com/questions/...
CommunityBot
- 1
5
votes
0
answers
133
views
Banach spaces admitting no proper quasi-affinity
I am interested in examples of Banach spaces $X$ satisfying the following two conditions:
(1) Every (continuous linear) injective operator $T:X\to X$ with dense range is surjective.
(2) $X$ is ...
- 4,461
5
votes
1
answer
378
views
Is $H^\infty$ a second dual space?
Let $H^\infty$ denote the Banach space of all bounded analytic functions on the open disc $\mathbb{D}$. It is easy to see that $H^\infty$ is a dual space. However, is there a Banach sapce $Y$ such ...
- 25.8k
5
votes
3
answers
1k
views
Sequential closure of a set: standard terminology, notation, and properties
Let $X$ be a topological vector space (or, perhaps, more generally uniform space). Let $A\subset X$ be a subset. Let $A^s$ denote the set of limits of all convergent sequences (I guess $A^s$ is called ...
CommunityBot
- 1
4
votes
0
answers
693
views
On the projective tensor product of $c_{0}$ by $c_{0}$
Let $E$ be the projective tensor product of $c_{0}$ by $c_{0}$. Does it follow that $E$ is isomorphic to no subspace of $C(K)$, where $K$ is countable compact metric space?
When $C(K)$ is isomorphic ...
- 81
1
vote
1
answer
237
views
Interpolation and embeddings for parabolic function spaces
I have a somewhat easy looking question on parabolic function spaces:
Let $B$ be a ball in $\mathbb R^n$ and let $T>0$. Denote $Q:=B \times [0,T]$. Assume $f \in L^2(Q) \cap L^\infty(0,T; L^q(B))$ ...
- 516
4
votes
1
answer
439
views
Characterization of $l_p$ up to a linear isometry
There is a science called "Geometry of Banach spaces". I wonder if they managed to give a geometric characterization of $\ell_p$ ($p\in[1,\infty]$) up to isometric isomorphism (among all Banach spaces)...
CommunityBot
- 1
6
votes
2
answers
405
views
$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$
Let $T$ be a linear operator acting on a finite-dimensional real or complex
vector space. As a direct consequence (or rather a particular case) of the
Riesz-Thorin theorem, we have
$$ \|T\|_2 \le \...
- 7,633
6
votes
1
answer
2k
views
Existence of injective operators with dense range
Given two separable (infinite dimensional) Banach spaces $X$ and $Y$, it is not difficult to show that there exists an injective (bounded linear) operator $T:X\to Y$ with range dense in $Y$. See S. ...
- 31.5k
13
votes
2
answers
768
views
How hard (P, NP, NP-hard) is it to compute Schur norms of matrices (as multipliers)?
Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space (...
- 28.6k
0
votes
0
answers
184
views
Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?
$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...
- 365
2
votes
0
answers
267
views
Banach space interpolation theory in terms of categories
I have recently learned a bit about higher category theory.
And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory?
Is it ...
- 25.8k
6
votes
0
answers
532
views
Products of spaces containing no copies of $\ell_2(\Gamma)$
Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products.
When $\Gamma$ is countable the answer is ...
- 4,461
6
votes
0
answers
175
views
Ultrapowers and complemented subspaces
Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers.
It is not difficult to see that if $Y$ is ...
- 11.3k
3
votes
1
answer
365
views
Final topology of surjective linear map on Banach space
Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...
- 16.4k
4
votes
1
answer
207
views
Introducing a dual space structure
Suppose that we have a Banach space $X$ together with some locally convex Hausdorff topology on $X$, weaker than the one given by the norm, which makes the unit ball of $X$ compact. Is $X$ (Banach-...
- 42.8k
3
votes
1
answer
255
views
A differentiable version of the Michael selection theorem
Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map.
Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?
- 356
1
vote
0
answers
127
views
Mixed Tsirelson Norm
A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...
CommunityBot
- 1
3
votes
1
answer
432
views
A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable
Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...
- 3,584
7
votes
1
answer
411
views
Banach spaces with no reflexive complemented subspaces
If $X$ is a Banach space with the Dunford Pettis Property (DPP), then no infinite reflexive subspace can be complemented. Suppose now that the Banach space has the property, that no infinite reflexive ...
- 11.3k
5
votes
1
answer
243
views
Complemented subspaces of ultrapowers
It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented ...
- 31.5k
5
votes
1
answer
808
views
Separable Banach spaces which are absolute Lipschitz retracts
A subset $F$ of a metric space $M$ is called a Lipschitz retract of $M$ if there is a Lipschitz map from $M$ onto $F$ which coincides with the identity on $F$. A metric space which is a Lipschitz ...
CommunityBot
- 1
9
votes
3
answers
4k
views
Projections in Banach spaces
Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
CommunityBot
- 1
6
votes
2
answers
548
views
When is it $C(X)$?
Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=...
- 4,461
2
votes
0
answers
132
views
Birkhoff orthogonal of a Banach space in its bidual
Let $X$ be a Banach space embedded in $X^{**}$ in the usual way.
We consider the set
$$
O_X := \{ x^{**}\in X^{**} : \|x^{**}-x\|\geq \|x\| \textrm{ for all }x\in X\}.
$$
I think this is the ...
- 4,461
6
votes
3
answers
1k
views
Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?
Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
- 29.8k
1
vote
1
answer
160
views
Contractively complemented subspaces without contractively complemented complement
Can someone give me an example of a Banach space $X$ and contractive projection $P\in\mathcal{B}(X)$ such that $\ker P$ is not a range of any contractive projection $Q\in\mathcal{B}(X)$?
CommunityBot
- 1
0
votes
2
answers
205
views
ANR Subsets of banach spaces
I need a reference for conditions on a closed subspace of a Banach space to have the homotopy type of an ANR.
CommunityBot
- 1
2
votes
1
answer
207
views
tensorial product with Lp
Let us consider E a finite-dimensional normed space on $\mathbb{R}$ and a real number $p\geq 1$.
Is it true that the projective tensorial product $E \widehat{\otimes}_\pi L^p(\mathbb{R},\mathbb{R})$ ...
- 1,697
5
votes
2
answers
606
views
Banach-Mazur distance to complex $\ell^1$ of a space containing real $\ell^1$
Consider a complex Banach space $X$ with a real subspace isometric to $\ell^1_{\mathbb R}$. What is the best constant $c$ such that $X$ contains a complex subspace $c$-isometric to $\ell^1_{\mathbb C}$...
- 31.5k
2
votes
1
answer
577
views
When is the bound in Riesz-Thorin Interpolation Theorem attained?
Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
- 13k
2
votes
2
answers
214
views
Representation of Banach spaces partially ordered by solid, normal, minihedral cones
I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989.
Theorem. Let $E$ be ...
- 15.5k
4
votes
1
answer
670
views
A generalization of a theorem of Grothendieck
In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$.
Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.
Assume that $S$ is a subvector space of ...
- 356
2
votes
1
answer
365
views
On sequences which converge to zero with respect to an operator ideal
Let $X$ be a Banach space and $\mathcal{A}$ be an operator ideal. A sequence $(x_{n})_{n=1}^{\infty}$ in $X$ is called $\mathcal{A}$-convergent to zero if there exist an operator $S\in \mathcal{A}(Z,X)...
