All Questions
1,222 questions
5
votes
2
answers
404
views
Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?
The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...
- 181
0
votes
0
answers
97
views
Is there any concise sufficient condition for the dual space to have Kadec property?
A normed space $E$ has a
Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.
Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also ...
- 5,529
8
votes
0
answers
384
views
What is the name for a Banach space property closed under ultraproducts?
In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (...
- 6,287
4
votes
0
answers
534
views
$L_\infty(\mu)$ spaces non-isomorphic to a dual space
Given a measurable space $(\Omega,\mu)$ such that $L_\infty(\mu)$ is isomorphic to a dual space, $L_\infty(\mu)$ is an injective Banach space. Indeed, given a subspace $Y$ of $X$ and a norm-one ...
- 4,461
5
votes
1
answer
396
views
What Approximation Property does the space of Schatten-p class operators have?
Background
This is a follow-up question to:
What (classes of) Banach spaces are known to have Schauder basis?
In the previous question, I asked about what spaces are known to have Schauder basis. It ...
- 365
7
votes
3
answers
1k
views
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 ...
- 29.8k
5
votes
0
answers
186
views
Norm of projection onto functions of mean zero
Let $X$ be a finite set and consider the space $\ell^2(X;Y)$ of functions $\zeta:X\to Y$, where $Y$ is a fixed Banach space. It decomposes into a direct sum of constant function and its complement $\...
- 51
1
vote
0
answers
109
views
Two tensor product norms inducing different topologies on the space of simple tensors
Are there two Normed spaces $V,W$ for which the algebraic tensor product $V\otimes W$ admits two different norms, both satisfying $\parallel x \otimes y \parallel= \parallel x \parallel. \...
- 356
5
votes
0
answers
204
views
quasi-weakly compact operators, co-ideals of operator ideals, and Banach spaces $X$ with $X^{**}/X$ separable
Throughout, $X$ and $Y$ will denote Banach spaces with $T\in\mathcal{L}(X,Y)$ (the space of continuous linear operators between $X$ and $Y$). We define the operator $\overline{T}\in\mathcal{L}(X^{**}/...
- 1,591
2
votes
0
answers
201
views
Reflexive subspaces of dual spaces
If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...
- 1,361
4
votes
1
answer
299
views
A question on $p$-approximation property
We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact ($1\leq p<\infty$) if there exists a $p$-summable sequence $(x_n)_{n=1}^{\infty}$ in $X$ such that
$$ K\subseteq \left\{\...
- 3,343
5
votes
0
answers
2k
views
Denseness of finite rank operators in $\mathcal{B}(X,Y)$
Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on
https://math.stackexchange.com/questions/...
- 107
2
votes
0
answers
111
views
proving that $\mathcal{A}_\infty(X)$ is or is not norm-closed in $\mathcal{L}(X)$ for each Banach space $X$
Fix any $1\leq p\leq\infty$. If $X$ is a Banach space and $C\in(0,\infty)$, we say that $T\in\mathcal{A}_C(X)$ whenever, for each $(x_n)_{n=1}^\infty\subset B_X$ (where $B_X$ is the closed unit ball ...
- 1,591
0
votes
1
answer
179
views
Dense subspaces of $L^p(0,T;X)$
Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert f\Vert_{X}^pdt<\...
- 1
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
5
votes
1
answer
298
views
If $ F(x,\bullet) \in {L^{\infty}}(G,B) $ for all $ x \in G $, then is $ x \mapsto F(x,\bullet) $ strongly measurable?
Let $ (X,\Sigma,\mu) $ be a $ \sigma $-finite measure space and $ B $ a Banach space. A function $ f: X \to B $ is said to be strongly $ \mu $-measurable iff it is the almost-everywhere pointwise ...
- 1,396
7
votes
2
answers
657
views
Subspaces isomorphic to $C[0, \omega_1]$
Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the ...
- 11.3k
0
votes
0
answers
123
views
On the operators from $l_{p}$ into Tsirelson's space $T$
Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
- 3,343
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
0
votes
1
answer
264
views
Banach space dual to $L^\infty(I,H^1(M))$
What is the dual to $L^\infty (I,H^1(M))$?, where $I$ is an interval in the real line; $H^1(M)$ is Sobolev space of degree 1, and $M$ is a compact manifold like the torus.
Any references that show ...
- 1,594
4
votes
0
answers
171
views
quasi-nilpotent part of a dual operator
Definitions and notation.
Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as
\begin{equation*}H_0(T):=\left\{...
- 1,591
7
votes
1
answer
1k
views
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?
- 8,512
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 ...
- 113
7
votes
1
answer
439
views
series representation in injective tensor products
All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product
$X \tilde{\otimes}_ \pi Y$ has a ...
- 16.4k
5
votes
0
answers
138
views
Banach spaces complemented in their ultrapowers
By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be ...
- 11.3k
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 ...
- 121
16
votes
0
answers
542
views
$C^*$-algebra generated by those operators that are bounded on every $\ell_p$
Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
- 25.8k
4
votes
1
answer
351
views
split of s.e.s. of Banach spaces
Let $l^{\infty}$ be the Banach space of all bounded real sequences with the $sup$-norm and $c$ the closed subspace of convergent sequences. Is there a continuous linear map $T: l^{\infty} \rightarrow ...
- 11.9k
1
vote
1
answer
71
views
Every open convex-valued multimap has global sections?
Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is ...
- 1,368
12
votes
3
answers
646
views
Radii and centers in Banach spaces
Suppose I have a Banach space $V$ and a set $A \subseteq V$ such that for all $\epsilon > 0$ there exists $v$ such that $A \subseteq \overline{B}(v, r + \epsilon)$. Does there exist $c$ such that $...
- 1,331
4
votes
2
answers
535
views
On hyperplanes of $L\infty$
Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is:
1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" ...
- 607
4
votes
0
answers
110
views
Banach space admitting a unique subsymmetric basis but not a symmetric one
I have two quick questions:
It can be shown without too much trouble (using methods from Altshuler/Casazza/Lin, 1973) that any Lorentz sequence space admits a unique (up to equivalence) subsymmetric ...
- 1,591
2
votes
0
answers
111
views
Ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$
I would like to know if there exist an explicit decription of ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1(\...
- 1,697
6
votes
0
answers
484
views
Square and cube?
Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...
- 113
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 ...
- 41
5
votes
0
answers
175
views
A Banach space with the BD property and without the weak Gelfand-Phillips property
A subset A of X is called Grothendieck if every operator T from X to $c_0$ maps A to a relatively weakly compact set.
A Banach space has the weak Gelfand-Phillips property (wGP) if every ...
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 ...
- 105
2
votes
1
answer
143
views
Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem
I am considering the following abstract Cauchy problem on Banach space $X$:
\begin{cases}
u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\
u(0)=x_0,
\end{cases}
Suppose $A$ generates an analyitc ...
- 503
6
votes
2
answers
2k
views
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.
- 71
4
votes
3
answers
3k
views
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 ...
- 8,512
0
votes
1
answer
2k
views
Infinite linear span vs closed linear span
Hi,
Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an ...
- 1,769
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)...
- 7,426
2
votes
1
answer
108
views
Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$
Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L_{p})$ and $l_{p}$-strictly singular operators, 2008, DOI: 10.4171/JEMS/141, eudml, arxiv) observed that if $(x_{n})_{n}$...
- 3,343
5
votes
1
answer
256
views
A Banach space with all Hilbertian subspaces complemeneted
Assume that $X$ is a Banach space in which every Hilbertian subspace is complemented (let's say that all the projections are uniformly bounded). What can we say about $X$? It has to be K-convex. By ...
- 5,005
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 ...
17
votes
0
answers
488
views
Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
4
votes
0
answers
209
views
On the weakly sequential completeness of the dual of the James space $J$
Let me first introduce some definitions. Let $1\leq p\leq \infty$.
A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is ...
- 3,343
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
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
154
views
An operator factoring through a Banach space containing no copy of $l_{1}$
Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_{1}$, but does not factor through any Banach space $W$ containg no copy of $l_{1}$?...
- 3,343