All Questions
1,222 questions
2
votes
1
answer
324
views
Characterization of nets with no convergent subnets in Banach spaces
Let $X$ be a finite-dimensional Banach space and $(x_i)_{i\in I}$ a net in $X$. Since every limited net in $X$ has a convergent subnet, it follows that $(x_i)_{i\in I}$ does not admits a convergent ...
- 38k
11
votes
1
answer
441
views
Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 31.5k
4
votes
0
answers
92
views
Simultaneous representations of elements of projective tensor products
Let $E,F$ be Banach spaces and consider the projective tensor product $E \widehat\otimes F$. If $\tau \in E \widehat\otimes F$ with $\|\tau\|<1$ then by definition we can find $(x_n)\subseteq E$ ...
- 18.7k
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
1
vote
1
answer
975
views
Annihilators and pre-annihilators
I asked this question on Math StackExchange first, but it was not answered.
If $X$ is a Banach space and $Z$ is a subset of $X^*$, consider the annihilator of $Z$ in $X^{**}$:
$$
Z^{\perp}=\{x^{**}...
- 4,878
2
votes
0
answers
115
views
Mean value of a map into Banach space
Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Let $(Y,\|\cdot\|)$ be a Banach space. Given a Bochner integrable map $f:X\to Y$ with $\|f\| \in L^2(X,\mu)$. The mean value of $f$ over $X$, ...
- 169
1
vote
1
answer
214
views
A generalization of strict convexity
Consider the following properties of a Banach space:
the intersection of any support hyperplane with the unit sphere is
(S) a singleton (this is the strict convexity);
(SF) finite-dimensional set;...
- 5,529
1
vote
1
answer
158
views
Extending functionals on $X^*$
Suppose $X$ is a non-reflexive Banach space, $Z$ a closed subspace of $X^*$, and $f$ a bounded functional on $Z$ with the property that there exists non-zero $x\in X$ such that $f(z^*)=z^*(x)$ for all ...
- 4,878
2
votes
0
answers
143
views
About a property of bounded closed convex set
Terminology:
For a bounded closed convex (bcc for short) set $A$, define $w(A)$ to be the infimum of the distance between pairs of
parallel hyperplanes supporting $A$.
We say that a bcc ...
- 31.5k
2
votes
1
answer
347
views
On complemented copy of $c_{0}$ in projective tensor products
Suppose that the projective tensor product of $l_{\infty}$ and $X$ contains a complemented copy of $c_{0}$. Does it follow that $X$ contains a complemented copy of $c_{0}$?
- 81
5
votes
1
answer
794
views
Can the Sobolev norm of order 1/2 detect "jumps"?
We are given a function $f: \mathbb R^d \to \mathbb R$. For simplicity we can assume that $f$ is smooth and compactly supported. Is the Sobolev norm of order $\frac{1}{2}$ strong enough to prove an ...
- 105
8
votes
1
answer
314
views
What algebras are quotients of $\ell_1(\mathbf{N})$?
Every separable Banach space is a linear quotient of $\ell_1$, however not every separable Banach algebra is a Banach-algebra quotient of $\ell_1(G)$ for some group $G$ (these are the so called ...
- 25.8k
3
votes
0
answers
169
views
A spanning set for an annihilator set on a Banach space
Let $(z_n)$ be a $H^\infty$-interpolating sequence on the open complex unit disc $\mathbb D$. If $A$ is some Banach space of analytic functions on the disc, denote by $X$ the closed subspace of all ...
- 31
9
votes
0
answers
261
views
SVD-type decomposition for the tensor product of three Hilbert spaces?
(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the ...
CommunityBot
- 1
3
votes
1
answer
222
views
Intuition for inequality involving permutation and Hamming Cube
Let $C^n=\{0,1\}^n$ be a metric space (Hamming Cube). The distance on $C^n$ is defined by
$$
d(\varepsilon,\varepsilon'):=|\{j:\varepsilon_j\ne\varepsilon'_j\}|,
$$
$\varepsilon=(\varepsilon_1,\...
- 1,245
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 462
1
vote
1
answer
190
views
Bounded operators on the Stinespring representation space
Let $A$ be a $C^*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed ...
- 18.7k
5
votes
2
answers
299
views
Banach space with an unconditional basis but not a quasi-greedy one?
A few years ago, Schechtman showed that $\ell_p(\ell_q)$ fails to admit a greedy basis whenever $1\leq p\neq q<\infty$. This furnishes an example of a Banach space with an unconditional basis but ...
- 1
0
votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 63.9k
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 ...
- 11.3k
1
vote
1
answer
221
views
Kernel of a convex combination of projections
Assume that we have finitely many projections $P_1,\dots, P_n$ on a Banach space $X$ (take an explicit case of $X=L_p(\mu)$ if a concrete examples is better). Consider their convex combination $P=\...
CommunityBot
- 1
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
4
votes
2
answers
543
views
Gaussian measure on Banach space
Assume we have a Gaussian measure $\mu$ supported on a Banach space $X$. Can we always find a Hilbert space $H$ embedded in $X$ sch that $\mu$ is also supported on $H$?
1
vote
0
answers
220
views
About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
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
1
vote
1
answer
109
views
Continuous factors for invertible simple tensors
Our following question is motivated by this very interesting answer
Assume that $A$ is a $C^{*}$ algebra. Put $X=\{a\otimes b \mid a,b \in G(A)\}$ where $G(A)$ is the space of all ...
CommunityBot
- 1
4
votes
2
answers
310
views
Geometric implications of $\beta(B_X) = 2$
Let $X$ be an infinite-dimensional Banach space and $\beta$ denote Istrățescu's spreading measure of noncompactness, i.e. $$\beta(M) = \sup \{ \varepsilon > 0 \colon \exists_{(x_n)^{\mathbb N} \in ...
- 446
3
votes
1
answer
204
views
Two questions on $l_{p}$-saturated Banach spaces
Let $1<p<\infty$. Recall that a Banach space $X$ is $l_{p}$-saturated if every infinite-dimensional subspace of $X$ contains a subspace isomorphic to $l_{p}$. I have a seemingly stronger notion ...
CommunityBot
- 1
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 4,713
5
votes
1
answer
109
views
Simultaneous near-best approximation with respect to two norms
Suppose that $M$ is a closed infinite dimensional subspace of $L_4(0,1)$ which is also a closed subspace of $L_1(0,1)$. Hence $M$ is isomorphic to $\ell_2$ as a subspace of $L_p(0,1)$ for $1\leq p\leq ...
- 23.7k
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
7
votes
2
answers
689
views
Which C*-algebras are complemented in their bidual?
Every von Neumann algebra is 1-complemented in its bidual, and so is every injective C*-algebra. Also, if $C_0(X)$ is infinite-dimensional and separable then it is not complemented in its bidual, and $...
CommunityBot
- 1
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
5
votes
1
answer
494
views
Is X+X finitely representable in X?
I wonder if the following assertion in true:
Conjecture. Let $X,Y,Z$ be infinite-dimensional Banach spaces such that both $Y$ and $Z$ are crudely finitely representable (c.f.r. for short) in $X$. ...
- 11.3k
4
votes
1
answer
215
views
almost invariant half space for a dual of a restricted operator
Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (...
CommunityBot
- 1
2
votes
1
answer
197
views
Explicit description of the closure of a given set
Let $C$ be the subset of $C_b(\mathbb{R})$ given by
$$C:=\{f\in C_b(\mathbb{R}):\ \exists f'\in C_b(\mathbb{R})\}$$
Now I want to take the closure of this set with respect to the supremum norm on $...
- 19.3k
1
vote
1
answer
158
views
When do we have $B_Y\subset T(B_X)$ if and only if $\overline{B_Y}\subset T(\overline{B_X})$?
Let $X$,$Y$ be normed spaces, $T:X\to Y$ be a bounded linear operator. Denote the open and closed unit balls by
$$
B_X:=\{ x\in X\ |\ \|x\|<1\} \\
\overline{B_X}:=\{ x\in X\ |\ \|x\|\le1\}
$$
and ...
CommunityBot
- 1
24
votes
2
answers
2k
views
Unique predual of a Banach space
Suppose $E$ is a dual Banach space whose predual is unique, and $E_0$ is a codimension 1 weak* closed subspace of $E$. Is the predual of $E_0$ necessarily unique?
Okay, I will reveal the motivation. ...
- 42.8k
2
votes
1
answer
238
views
Hilbert-irreducible Banach space
A Banach space $X$ is called Hilbert-irreducible if it satisfies the following condition:
If a subspace $Y\subset X$ satisfies the parallelogram equality, then $Y$ is necessarilly a one ...
- 54.2k
3
votes
1
answer
154
views
Uncomplemented copies of $\ell_2$ in $L_p(0,1)$ for $2-\varepsilon<p<2$
It was proved in [Bennett, G.; Dor, L.E.; Goodman, V.; Johnson, W.B.; Newman, C.M. On uncomplemented subspaces of $L_p$, $1<p<2$. Israel J. Math. 26 (1977), 178–187]
that, for $1 <p < 2$, ...
- 31.5k
5
votes
2
answers
1k
views
Are bounded sets always weakly metrizable in reflexive separable spaces?
It is known that if a Banach space is reflexive and separable, its unit ball is weakly metrizable.
My question is about the generalization of this property :
1) Is it true that for all reflexive ...
- 16.5k
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
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 ...
- 11.6k
4
votes
1
answer
222
views
If $K$ is a countable compact metric space is the set of extreme point of $Ba(C(K))$ countable?
The question is the title. The set $Ba(C(K))$ is the unit ball of $C(K)$. This has to be known, but I can't find the answer explicitly in the literature. There is some literature about polyhedral ...
- 109k
4
votes
1
answer
318
views
Non-equivalence of admitting different types of bases in Banach spaces
Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...
- 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
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 6,045
2
votes
0
answers
184
views
Properties of the optimal decomposition for the $K$-functional between $\ell_1$ and $\ell_2$
Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$:
$$
\lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...
- 1,372
2
votes
1
answer
292
views
Fixed point theorem for a nonconvex set in a Banach space
Generally speaking, I am looking for a generalization of the Schauder fixed point theorem, which applies to the situation described briefly below.
All references I read (e.g. E. Zeider 'Nonlinear ...
- 31.2k
2
votes
0
answers
346
views
When is the sum of complemented subspaces complemented?
Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
- 45.7k