All Questions
Tagged with fa.functional-analysis banach-spaces
1,222 questions
7
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Inductive/Projective Limits of Topological Algebras
It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...
- 71
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-...
- 744
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0
answers
294
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Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
- 175
7
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0
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124
views
The bidual of the space of divergence-free vector fields
Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
- 11.3k
7
votes
0
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248
views
Isometries on the unit sphere
Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
- 1,361
7
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0
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177
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Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?
I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
- 25.8k
7
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0
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327
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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
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0
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200
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Equivalent strictly convex norms in spaces of small density
Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm?
Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a ...
- 11.3k
7
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0
answers
183
views
Is there a quotient of $c_0$ without the approximation property?
The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...
- 71
7
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0
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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
7
votes
0
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293
views
Complex interpolation of a Banach space and its antidual when the space has a basis
Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
- 4,461
7
votes
0
answers
708
views
Minkowski's Inequality for Integrals in Orlicz spaces
EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.
Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$...
- 882
7
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0
answers
4k
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Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]
Possible Duplicate:
What’s an example of a space that needs the Hahn-Banach Theorem?
It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, say). ...
- 25.6k
6
votes
3
answers
2k
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reflexive banach space
I want to ask this non-expert question:
What does it mean geometrically for a Banach space to be reflexive?
Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
- 351
6
votes
3
answers
3k
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Why isn't the theorem of approximation applicable in Banach spaces?
Let X be a Hilbert space, A a convex, closed subset of X. Then there exists for every x in X exactly one best approximation in A, that is there exists a y in A such that || x-y || = d(x,A) = inf { || ...
- 61
6
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2
answers
4k
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Question about Schauder bases in C([0,1]).
I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:
The family of monomials $\{...
- 26.8k
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
6
votes
3
answers
427
views
Point-wise limit of finite valued functions
Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f_n\colon X\to X$ converging point-wise to the identity mapping on $X$?
- 4,058
6
votes
3
answers
2k
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Space of compact operators
I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...
- 11.3k
6
votes
3
answers
3k
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Non-empty resolvent set, then operator closed?
On Hilbert spaces, the following is true:
Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.
The obvious proof I see to show this uses explicitly the ...
- 115
6
votes
2
answers
749
views
Transpose of unbounded operators between Banach spaces.
Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator
$L' : \operatorname{...
- 5,327
6
votes
2
answers
1k
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Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable
... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...
- 25.8k
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})=...
- 306
6
votes
1
answer
618
views
Whether Krein-Milman property implies Radon-Nikodym property
A Banach space is said to have Krein-Milman property (KMP in short) if every closed bounded convex set of it is a closed convex hull of its extreme points. Eg. Any reflexive space has KMP, $\ell_1$ ...
- 521
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
6
votes
2
answers
735
views
Tensor product space with projective norm is incomplete
Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are ...
- 163
6
votes
2
answers
3k
views
Closed convex bounded sets are weakly compact for which spaces?
It is known that for all reflexive Banach spaces, closed convex bounded sets are weakly compact (compact for the weak topology).
What is the general class of topological vector spaces for which this ...
- 549
6
votes
2
answers
282
views
The Calkin representation for Banach spaces
Let $X$ be an infinite dimensional Banach space. Let $\Lambda_{0}$ be the set of all finite dimensional subspaces of $X$ directed by the inclusion $\subseteq$. For each $\alpha\in \Lambda_{0}$, let $...
- 3,343
6
votes
3
answers
2k
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Generalized Hardy-Littlewood-Sobolev Inequality
The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb R^n),\...
- 16.2k
6
votes
3
answers
1k
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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 ...
- 125
6
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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 ) ...
- 18.7k
6
votes
2
answers
3k
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Dense inclusions of Banach spaces and their duals
This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
- 8,512
6
votes
1
answer
381
views
Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?
Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$
is not ...
- 395
6
votes
1
answer
826
views
Non-reflexive Banach space s.t. X,X*,X**,... are separable
Is there an infinite-dimensional Banach space $X$, which is not reflexive, such that all the spaces $X,X^{\ast},X^{\ast\ast}, X^{\ast\ast\ast},\dots$ are separable?
- 11.9k
6
votes
1
answer
353
views
Sequential continuity of linear operators
Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces.
Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones.
(This notion is ...
- 21.8k
6
votes
2
answers
426
views
Ultrapowers of operators
Can we prove that for each infinite dimensional Banach space $X$ and any free ultrafilter (possibly over uncountable set of indices) $\mathcal{U}$ the obvious embedding
$$({\mathcal{L}(X)})_{\mathcal{...
- 143
6
votes
1
answer
335
views
Existence of pairwise quasi-complementary but not complementary subspaces
Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
- 391
6
votes
2
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458
views
Subspaces of $\ell_p$ ($1<p<\infty$, $p\neq 2$) not isomorphic to $\ell_p$
Is it possible to show the existence of an infinite dimensional closed subspace of $\ell_p$ ($1<p<\infty$, $p\neq 2$), not isomorphic to $\ell_p$, in an elementary way?
For $1<p<q<2$,...
- 4,461
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
6
votes
1
answer
237
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Radon-Nikodym property in Diestel & Uhl: a definition clarification
I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly ...
- 201
6
votes
1
answer
267
views
Unconditionally convergent series in $\ell_2$ consisting of $\ell_1$-small vectors
For a function $x:\omega\to\mathbb R$ let $|x|$ denote the function $|x|:\omega\to[0,\infty)$, $|x|:n\mapsto|x(n)|$.
It is well-know that a series $\sum_{n\in\omega}r_n$ of real numbers converges ...
- 41.8k
6
votes
2
answers
509
views
A question on Grothendieck space
A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck ...
- 3,343
6
votes
1
answer
377
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quick question about renorming quasi-Banach spaces into p-Banach spaces
I have a quick question which is probably supposed to be obvious, but for some reason I just don't see it: How does one re-norm a quasi-Banach space to produce a $p$-Banach space ($0<p\leq 1$) ...
- 1,591
6
votes
1
answer
450
views
Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?
A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...
- 41.8k
6
votes
1
answer
238
views
Extending a weak*-converging sequence onto a superspace
Let $X$ be a real Banach space and $Y\subset X$ be a (closed) subspace of $X$. Assume that a sequence $y_n^*\in S_{Y^*}$ weak*-converges to some $y^*\in S_{Y*}$. (Here $S_{Y^*}$ stands for the dual ...
6
votes
1
answer
223
views
Is the space of vectorial functions that are Dunford integrable complete?
Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...
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 \...
- 23k
6
votes
1
answer
364
views
A question about uniformly bounded semigroups
Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there ...
6
votes
1
answer
2k
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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. ...
- 4,461
6
votes
1
answer
273
views
Quasi-reflexive spaces which are not isometric to dual spaces
My question may sound weird and I have no deep motivation behind it other than curiosity.
As is well-known, quasi-reflexive spaces have the Radon-Nikodym property hence their balls have lots of ...
