All Questions
1,222 questions
4
votes
1
answer
428
views
Reference Request: Calculus of Variations in Hilbert Space
I'm looking for a good reference to a book on calculus of variations in the setting of Banach Spaces.
If it helps, I'm working with a particular functional acting on Fr\'{e}chet-differentiable ...
3
votes
1
answer
177
views
Rate of convergence of weakly null sequences
If $x_n$ is a normalized, weakly-null sequence in a Banach space, and $\epsilon_n\to 0$, does there exists a non-zero functional $f$ such that $|f(x_n)|<\epsilon_n$ for all $n$?
- 1,361
12
votes
1
answer
191
views
Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 305
13
votes
0
answers
325
views
Banach spaces with $d(X,Y) = 1$
We recall that the Banach-Mazur distance between two isomorphic Banach spaces is given by $d(X,Y) = \inf \{ \|T\| \|T^{-1}\| : T$ is an isomorphism from $X$ to $Y\}$.
It is a classical result that we ...
- 538
2
votes
2
answers
374
views
A criterion for norming sets
Let $F$ be a Banach space with the closed unit ball $B$. Let $E\subset F^*$ be a total subspace such that $B$ is complete with respect to the norm $|||f|||=\sup \limits_{e\in E,~e\ne 0} \frac{|\left&...
- 5,529
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
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 ...
- 1,591
7
votes
1
answer
556
views
Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC
Let's note $E=C([0,1],\mathbb{R})$ the Banach space of real continuous funtions from the [0,1] interval with the uniform norm.
Is it possible to show a non-continuous linear form on $E$ exists ...
- 1,355
5
votes
4
answers
1k
views
Is any continuous linear operator from a dual Banach space to a separable Hilbert space the strong-operator limit of a net of adjoint operators of less or equal norm ?
Let $E$ be an arbitrary Banach space and let $T:E^{*}\rightarrow\ell^{2}$
be a linear continuous operator. Is it true that $T$ must be the
$so$-limit (i.e., limit w.r.t. the strong operator topology) ...
- 4,060
10
votes
2
answers
881
views
volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
- 1,222
5
votes
1
answer
602
views
Invariant probability on a unit ball of a Banach space
Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...
- 53
5
votes
2
answers
296
views
Well-complemented copies of $\ell_p^n$
This must be surely known but I couldn't locate this problem in the literature. It popped out in a priori unrelated approximation problem but if true, would help me greatly.
Let $p\in (1,\infty)$.
...
- 11.3k
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
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 $H$?
- 43
-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
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 ...
- 161
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 $...
16
votes
1
answer
2k
views
What (classes of) Banach spaces are known to have Schauder basis?
Motivation:
I am trying to see for what class of Banach spaces the following result is true:
There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
- 365
1
vote
0
answers
27
views
Approximation of multipliers by multipliers of a smaller set 2
This question is a refinement of my previous question.
Let $X$ be a compact metric space, and let $B$ be a bounded Banach Disk in $C(X)$ such that for every $x\in X$ there is $f\in B$ with $f(x)\ne 0$...
- 5,529
5
votes
0
answers
245
views
Examples of Banach lattices with positive Schur property but without Schur property
A Banach lattice $E$ has the
$(1)$ Schur property provided that any weakly null sequence in $E$ is norm null in $E$, and
$(2)$ positive Schur property provided that any weakly null sequence of ...
user114263
2
votes
1
answer
119
views
Finite-representability of $\ell_p$ in super-reflexive spaces
Let $E$ be a Banach space. Is it possible that $E$ is super-reflexive and $\ell_p$ is crudely finitely representable in $E$ for all $p\in (1,2)$?
It seems unlikely but I cannot find an argument off ...
- 331
10
votes
1
answer
515
views
Complemented subspaces in the dual of James' space $J$
James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...
- 4,461
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
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
1
vote
1
answer
220
views
Criterion of reflexivity
Let $E$ be a Banach space.
It is known that if for any equivalent norm on $E^*$ the closed unit ball of $E^*$ is weakly* closed, then $E$ is reflexive (a very short proof is in the book by Fabian, ...
- 5,529
2
votes
1
answer
746
views
A unital algebra with norm and continuous multiplication is a Banach algebra
In my research in functional analysis, I came across this rather simple result:
For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication , right and left are both continuous w....
- 221
4
votes
1
answer
254
views
M-bases for $C(K)$-spaces, $K$ -scattered
Recall that a biorthogonal system $\{(x_i, x^*_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x_i\}_{i\in I}$ is linearly dense in $E$ and $\{x_i^*\}_{i\in I}$ separates points. Let me ...
- 11.3k
4
votes
0
answers
212
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)...
3
votes
1
answer
807
views
Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?
Question 1 Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?
Clearly if $Y$ is closed in the norm topology ...
- 243
3
votes
1
answer
438
views
Separable subspaces in dual spaces
Let $X$ be a Banach space and $Y$ be a separable closed subspace of $X^{*}$. Is there a separable closed subspace $Z$ of $X$ such that $Y$ is isomorphic to a subspace of $Z^{*}$? Thank you!
- 3,343
0
votes
1
answer
218
views
Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
- 167
3
votes
2
answers
278
views
Centralizers and containment of $c_0$
I have this question also in MSE (see: https://math.stackexchange.com/questions/666053/centralizers-and-containment-of-c-0), but I have not got an answer there. So I thought I try my luck here.
Let $...
- 125
8
votes
2
answers
1k
views
When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?
I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive ...
12
votes
1
answer
306
views
Containment of $c_0$
I have the following question. I guess it's quite simple for experts.
Unfortunately, I could not come up with an answer yet.
Let $X$ be a Banach space which contains no copy of $c_0$.
Does it impply ...
- 225
9
votes
1
answer
2k
views
Dual or pre-dual of BV
Was there any relevant work to determine the dual (or more likely the predual) of the space of bounded variation functions $BV(\mathbb{R}^n)$ (I recall the definition : a function in $L^1(\mathbb{R}^n)...
- 1,616
0
votes
1
answer
136
views
When are Weighted $\mathcal{L}^p$-Spaces Topologically Isomorphic?
Let $X$ be a topological space and $\mu$ be the Borel measure on $X$. Suppose $W_1$ and $W_2$ are continuous, non-negative functions from $X$ into the real numbers such that, for all integers $p > ...
- 263
4
votes
1
answer
498
views
Generator of a $C_0$-semigroup restricted to a subspace
Suppose we have a decreasing filtration of Banach spaces $(E_t)_{t\geq0}$, inclusions $V_{s+t,t}:E_{s+t}\to E_t$ and projections $P_{t,s+t}:E_t\to E_{s+t}$ such that $P_{t,s+t}V_{s+t,t}=I_{E_{s+t}}$. ...
- 1,411
5
votes
1
answer
237
views
Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?
More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...
- 3,584
16
votes
2
answers
4k
views
Usefulness of Frechet versus Gateaux differentiability or something in between.
If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux ...
- 943
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 ...
- 1,361
2
votes
1
answer
167
views
Distortion of embedding in Hilbert space
Given an injective linear map $T$ between Banach spaces $X$ and $Y$, let
\begin{equation} d(T) = \sup \left \{ \frac{||x||_X}{||Tx||_Y}: x \in X \mbox{ is nonzero } \right\} \cdot ||T||_{\mathrm{op}}...
- 1,769
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
1
vote
1
answer
96
views
About representations of some elements in $\mathcal A(\ell^p)$
For Banach spaces $E$ and $F$ we denote the approximate operators by $\mathcal A(E,F)$ and projective tensor product by $\hat\otimes$.
Consider the natural map
$$\Delta: \mathcal A(\ell^q,\ell^p)\...
- 2,118
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
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
0
answers
552
views
Normed space between $H^{0+}$ and $L^2$
Cosider a function $f\in L^2(\mathbb{R}^3)$ with consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty \, .\qquad\mbox{(*)} \, $$
Of course if $f\in ...
- 943
1
vote
0
answers
271
views
When Pelczynski's property (V*) forces (V) in the dual space?
Recently, I have been asked if are there any sufficient conditions for a Banach space with Pelczynski property (V*) to have dual with property (V). Since it was a long standing open problem, I guess ...
2
votes
0
answers
189
views
Dunford−Pettis property of $L^1(\mu)$
$\def\bs#1{\boldsymbol#1}\def\sp{\kern.4mm}$Let $\bs K$ be either the standard real or complex topological field, and let $E$ be a Hausdorff locally convex space over $\bs K\sp$. Then saying that $E$ ...
- 3,584
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
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