All Questions
1,222 questions
7
votes
1
answer
291
views
Does separability of the strong operator topology imply separability of the underlying space?
Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$.
Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum.
...
- 4,058
8
votes
2
answers
488
views
If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?
- 4,058
2
votes
1
answer
106
views
Norming functionals for vectors in intersections
Suppose that $(X, \|\cdot\|_X)$, $(Y, \|\cdot\|_Y)$ are two Banach spaces such that $X\subset Y$ and $\|x\|_Y\leq \|x\|_X$ for all $x\in X$ and $X$ is dense in $(Y, \|\cdot\|_Y)$.
Every functional $...
- 23
2
votes
1
answer
230
views
Relation between the weak star topology and hereditary Lindelöfness
Let $X$ be a Banach space. Is the following implication valid?
$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$
The converse is clearly true, since the ...
- 4,058
8
votes
1
answer
232
views
Lipschitz right inverses of Banach space quotients
Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear).
Of course, if $Y$ is ...
4
votes
2
answers
336
views
pointwise convergence to the identity
Let $X$ be a separable topological vector space with size (cardinal number) no larger than $\mathfrak{c}$. Does there exist any sequence of finite rank linear maps $\phi_n:X\to X$ pointwise converging ...
- 4,058
4
votes
0
answers
115
views
point-wise approximation of the identity in hereditary Lindelof spaces
Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.
Q. Can we concluded that $X$ is hereditery ...
- 4,058
8
votes
1
answer
305
views
Subspaces isomorphic with quotients
Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?
- 1,361
4
votes
0
answers
166
views
Is this property an isomorphic characterization of $\ell_1(\Gamma)$?
Let $\Gamma$ be an infinite set. Then every $(x_i)_{i\in\Gamma}\in \ell_1(\Gamma)$ has at most a countable number of components $x_i\neq 0$.
As a consequence, every separable subspace $M$ of $\ell_1(\...
- 4,461
10
votes
1
answer
366
views
Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 41.9k
1
vote
0
answers
217
views
Status of an open problem in isometric aspect of Banach space theory
The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$
Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach ...
- 623
2
votes
3
answers
3k
views
$c_0$ is not isometrically isomorphic to $c$
Let us consider the space of convergent sequences which is denoted by $c$. The space of all sequences $(x_n)\in c$ with $\lim x_n=0$ is also denoted by $c_0$. Clearly $c_0$ is a proper closed ...
- 4,058
2
votes
1
answer
151
views
A particular separation example
Q1. Does there exist a separable Banach space $X$ satisfying in the following property?
1- $X^*$ is non separable.
2- For every countable subset $F\subset X^*$ there exists $0\neq x_F\in X$ ...
- 4,058
5
votes
1
answer
197
views
The largest topological copy of a Hilbert space contained in $\ell^1$
Let us consider $\ell^1$, the space of absolutely summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$.
...
- 4,058
9
votes
1
answer
384
views
Comparing two $\sigma$-algebras on $B(\ell^1)$
Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow
$$w-\lim T_i=T \Longleftrightarrow \...
- 4,058
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
4
votes
1
answer
394
views
Separable Lindelöf locally convex spaces that are not second-countable
A Lindelöf space is a topological space in which every open cover has a countable subcover.
Does there exists a Lindelöf locally convex space which is not second countable?
I am also looking for a ...
- 4,058
11
votes
3
answers
2k
views
Is the strong operator topology metrizable?
Let $X$ be a separable Banach space. Is the strong operator topology metrizable on $B(X)$, the space of all bounded operators on $X$?
SOT-$\lim T_i=0~$ if and only if $~\lim \|T_ix\|=0$ for every $x\...
- 4,058
7
votes
0
answers
328
views
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
0
votes
0
answers
161
views
Topologies corresponding to norm, SOT and WOT under duality
This is a question from MSE which has not received any attention so far.
Let $X$ be a Banach space with norm dual $X'$. (I am mostly interested in the case $X = \ell^1$.)
For a linear mapping $T : X \...
- 1,773
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$.
...
- 41.9k
3
votes
0
answers
109
views
Is the Banach space $C(K)$ a $1$-Lipschitz comp-extensor?
Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map $...
- 41.9k
8
votes
1
answer
361
views
What is the smallest Lipschitz constant of a Lipschitz retraction of $\ell_\infty([0,1])$ onto $C[0,1]$?
By Theorem 1.6 in the book "Geometric Nonlinear Functional Analysis" by Benyamini and Lindenstrauss, the Banach space $C[0,1]$ is a Lipschitz retract of the Banach space $\ell_\infty[0,1]$. ...
- 41.9k
8
votes
1
answer
325
views
Why $S$ cannot be homeomorphic to the $1$-sphere of $\ell^2$?
Consider the $\ell^2$ complex Hilbert space.
Let $m\in \mathbb{N}^*$ be a fixed number, and set
$$
S=\left\{ x=(x_n)_n\subset \ell^2\ :\ \sum_{n=1}^m \frac{|x_n|^2}{n^2}=1\right\}.$$
I want to ...
- 724
7
votes
0
answers
200
views
Equivalent strictly convex norms in spaces of small density
Can one construct in ZFC a Banach space of density character $\omega_1$ that does not have an equivalent strictly convex norm?
Maybe one may apply some kind of a Löwenheim–Skolem-type argument to a ...
- 11.3k
2
votes
1
answer
96
views
A Question about an irreducible ultra-power II,
Let $E$ be an irreducible Banach $A$-module, for a Banach algebra $A$. One can easily show that for an ultra filter $\mathcal U$, $(E)_\mathcal U$ is a Banach $(A)_\mathcal U$-module. Is it possible ...
- 2,118
8
votes
1
answer
268
views
Two questions about basic sequences
Suppose $(x_n)$ and $(y_n)$ are two basic sequences in a separable Banach space $X$ such that $\overline{span}\{(x_n), (y_n)\}=X$. Can we always pass to subsequences $(x_{n_k})$ and $(y_{n_k})$ such ...
- 355
0
votes
0
answers
977
views
Weak convergence can imply strong convergence [duplicate]
In $\ell^1(\mathbb N)$, weak convergence implies strong convergence. Is there a classification of infinite-dimensional Banach spaces for which such a property holds true ?
- 16.2k
2
votes
0
answers
149
views
Projection semigroup of an isolated eigenvalue
I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
- 381
2
votes
0
answers
124
views
Logarithm of $L^p$ space
I encountered the following space as a natural space for setting up a certain problem:
$$
S_m^p = \{f \colon I \to \mathbb{R} \text{ measurable }; m^{f} \in L^p(I)\}
$$
Here, $I$ is an open bounded ...
- 648
0
votes
0
answers
70
views
A question about an irreducible ultra-power
Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
- 2,118
2
votes
1
answer
185
views
Almost homogeneous functions
Let $X$ and $Y$ be Banach spaces and $T: X \to Y$. Working with large scale geometry of Banach spaces, I reached the following property:
Suppose that for every scalar $\alpha\in\mathbb K$ and every ...
- 563
2
votes
0
answers
199
views
Uniformly convex, uniformly smooth Banach space which is not convex of power type
It is well known that every uniformly convex Banach space $X$ admits an equivalent norm which such that it is convex of power type, i.e. the modulus of convexity with respect to the new norm satisfies ...
- 799
14
votes
4
answers
550
views
About the existence of characters on $B(X)$
Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M_n(\mathbb{C})$ ...
- 355
4
votes
0
answers
211
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
-1
votes
1
answer
132
views
About a property in a reflexive Banach space
Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
- 2,118
7
votes
2
answers
573
views
Existence of spectral gap
I would like to start by saying that any comment or idea is highly appreciated.
Let us observe that for Hilbert-Schmidt operators $H_1,H_2$ on an infinite-dimensional separable complex Hilbert space $...
- 95
7
votes
1
answer
305
views
Reflexive subspaces of bidual Banach spaces
The answer to the question is almost surely negative (as almost always in Banach space theory) but I cannot find a relevant example.
Is there an example of an infinite-dimensional Banach space $X$ ...
- 75
5
votes
0
answers
134
views
Banach space properties defined by compact operators, strictly singular operators and strictly cosingular operators
Let $X,Y$ be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from $X$ into $Y$, $\mathcal{K}(X,Y)$ by the space of all the compact operators from $X$ into $Y$, $S(X,Y)$ by ...
- 3,343
2
votes
1
answer
387
views
The closure of span of a linearly independent and convergent sequence in $\ell^2$ [closed]
Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to v_0$.
I would like to know if ...
- 433
10
votes
0
answers
251
views
Do sufficiently large Banach spaces admit non-compact operators with not too large range?
As in the title,
does there exist a cardinal number $\lambda$ such that for every Banach space $X$ of density/cardinality at least $\lambda$ there exists a non-compact bounded, linear operator $T\...
- 11.3k
1
vote
1
answer
268
views
About reflexivity of ultrapower
It is obvious that for a Banach space $E$, $E$ is reflexive iff $\ell^2(E)$ is reflexive. Let $\mathcal U$ be an ultrafilter. Is the reflexivity of $(E)_\mathcal U$ equivalent to refelxivity of $(\ell^...
- 2,118
0
votes
0
answers
55
views
Continuity of a composite function
Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.
Is the mapping
\begin{equation}
\begin{array}{rcl}
C^0([0,T],C^1(\bar{\...
- 143
5
votes
1
answer
1k
views
Reference request: The resolvent is analytic in the resolvent set
I am busy reading through Taylor's paper Spectral Theory of Closed Distributive Operators.
On page 192, he defines the resolvent and spectrum of $T$:
Later on in the paragraph, he then proceeds by ...
- 289
2
votes
0
answers
150
views
Non-separable asymptotic $\ell_1$ space
The Figiel-Johnson Tsirelson space is an example of an asymptotic $\ell_1$ Banach space not containing $\ell_1$. The notion of asymptotic $\ell_1$ is with respect to some basis, but a coordinate free ...
user114263
3
votes
0
answers
149
views
Classical subspaces of non-atomic Banach lattices
Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $...
user114263
13
votes
1
answer
724
views
Trace-class operator satisfies $\sum |\lambda_n|<\infty$?
Here's an "exercise" which I thought should be easy, but which I find myself unable to do.
Let $V$ be a Banach space.
Recall that an operator $f:V\to V$ is trace-class if it is in the image of the ...
- 43.2k
6
votes
1
answer
212
views
Nice S¹-action implies existence of unconditional basis?
Let $V$ be a Banach space equipped with a continuous linear action of $S^1$ (meaning, the map $S^1\times V\to V$ is continuous). Assume that all the eigenspaces of the $S^1$-action are finite ...
- 43.2k
2
votes
0
answers
93
views
Open problems concerning Araujo's biseparating maps
Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$
Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
- 623
13
votes
2
answers
653
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 536