All Questions
1,222 questions
4
votes
2
answers
434
views
A homeomorphism between the unit interval $[0,1]$ and a linearly independent subset of a Hilbert space
Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$
Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ ...
- 63.9k
1
vote
1
answer
200
views
The intersection of closure of span of infinite, linearly independent, closed, bounded, separated subsets of $\ell^2$
Let $X$ and $Y$ be two subsets of $\ell^2$ space over $\mathbb{C}$ such that: $X \cup Y$ is linearly independent, $X \cap Y = \emptyset$ and $\inf_{x \in X, y \in Y} \| x-y \|>0$ and such that each ...
- 63.9k
5
votes
1
answer
1k
views
Space of compact operators defined on separable Hilbert space
Let $X$ be a separable Banach space and consider $\mathcal{K}(X)$ the space of compact operators $K\colon X \rightarrow X$. Is it true that the space $\mathcal{K}(X)$ is separable? If yes, why? If no, ...
- 11.3k
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
7
votes
1
answer
814
views
An equivalent condition for separability of $X^*$
Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms:
$$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$
...
- 18.7k
13
votes
0
answers
324
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 ...
- 63.9k
5
votes
1
answer
353
views
A dense subset in $B(X)$ under the weak operator topology
Let $X$ be a Banach space and consider $B(X)$, the set of all bounded linear maps on $X$. By the W-topology on $B(X)$ we mean the topology induced by the semi-norms
$$B(X)\to [0,\infty): T\to |\...
- 60.6k
4
votes
1
answer
104
views
Two locally convex topologies on $B(X)$.
Let $X$ be a non-reflexive Banach space. It is supposed to compare two locally convex topologies on $B(X)$:
Let $w$ be the topology on $B(X)$ implemented by all seminorms given by
$$B(X)\to [0,\...
- 18.7k
5
votes
0
answers
103
views
Complementation problem for $\ell_p^2$
Let $n\in\mathbb{N}$ and $p,q\in(1,+\infty)$ with $p^{-1}+q^{-1}=1$. Consider isometric embedding between $\mathbb{C}$-Banach spaces
$$
\rho:\ell_p^n\to\ell_\infty(S, \ell_1^n),x\mapsto(f\cdot x)_{f\...
- 1,697
11
votes
1
answer
953
views
Separable bidual but nonseparable third dual
Does there exist a Banach space $X$ such that $X^{**}$ is separable but $X^{***}$ is non-separable?
More generally, for every natural $n$ can someone construct an example of Banach space $X$ such ...
- 31.5k
1
vote
1
answer
791
views
$\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space
Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:
$X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
$X' \cong \ell_\infty ...
- 60.6k
5
votes
2
answers
853
views
Covering compactness in the weak sequential topology
Let $X$ be a real Banach space. Apart from the norm topology, we can consider the following weak topologies on $X$:
the weak toplogy, defined as the initial topology with respect to $X^*$. In other ...
- 171
4
votes
1
answer
428
views
Reference Request: Calculus of Variations in Hilbert Space
I'm looking for a good reference to a book on calculus of variations in the setting of Banach Spaces.
If it helps, I'm working with a particular functional acting on Fr\'{e}chet-differentiable ...
CommunityBot
- 1
1
vote
1
answer
96
views
About representations of some elements in $\mathcal A(\ell^p)$
For Banach spaces $E$ and $F$ we denote the approximate operators by $\mathcal A(E,F)$ and projective tensor product by $\hat\otimes$.
Consider the natural map
$$\Delta: \mathcal A(\ell^q,\ell^p)\...
- 2,118
3
votes
1
answer
200
views
A question on ultraproducts of $L_{p}(\mu)$-spaces
Let $I$ be a set, $\mathcal{U}$ be an ultrafilter on $I$ and $1\leq p<\infty$. Let $X_{i}=L_{p}(\mu_{i})(i\in I)$, where $\mu_{i}$ is a probability measure for each $i\in I$. Relying on standard ...
- 11.3k
7
votes
3
answers
754
views
Duality between Banach spaces and compact convex spaces
I always had the impression that there was a duality (i.e. a contravariant equivalence of categories) between Banach spaces and certain notion of pointed compact convex set (something like algebras ...
- 42.4k
2
votes
1
answer
138
views
Some questions on parabolic function spaces
I remember I read those problems some place, but I cannot find it. Does anyone have any idea where I can find it?
If $X$ is a Banach space, then $(L^1(a,b;X))^*\cong L^\infty(a,b;X^*)$?
$X, Y$ are ...
- 3,574
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 ...
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) ...
- 1,134
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 ...
CommunityBot
- 1
9
votes
3
answers
684
views
Lipschitz-free spaces of $\mathbb R^n$
We define
$$
\text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and }
\sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty.
\}
$$
It is well-known ...
- 11.3k
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 ...
- 17.2k
1
vote
0
answers
198
views
Morrey space is Banach space
I'm working with Morrey spaces, which are the spaces
$$L^{p,\lambda}(\Omega):= \left\{ u \in L^1_{loc}(\Omega): \sup_{x \in \Omega, r > 0} r^{-\lambda}\int_{B(x,r)\cap \Omega}|u(y)|^pdy< \infty\...
- 121
5
votes
2
answers
437
views
Sets in constructive mathematics
It is not completely clear how Bridges, Richman and Youchuan treated sets in their paper. Example is in the following lemma (Lemma 7 on p. 7):
Let $U$ and $V$ be (inhabited to mean $\exists u \in U, \...
CommunityBot
- 1
1
vote
1
answer
194
views
Proof that the subspace of signed measures integrating d(x,e) is closed
Let $\mathcal{M}(S)$ be a space of finite signed measures on a metric space $S$ ($=\mathbb{R}^2$ in my case) equipped with the total variation norm. Let
$\mathcal{M}_1(S)=\{\mu \in \mathcal{M}(S):\...
- 13
1
vote
1
answer
183
views
Criterion of reflexivity 2
Originally I meant to ask this question here, but got confused and ended up asking another question, which had some mathematical meaning, but was not what I vaguely had in mind.
Let me restate the ...
- 31.5k
5
votes
1
answer
519
views
Hahn Banach type extension of a Lipschitz map
The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by ...
- 101
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}}...
- 31.5k
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 ...
- 31.5k
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, ...
- 31.5k
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
14
votes
0
answers
205
views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
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&...
- 4,878
4
votes
0
answers
144
views
Embedding of $\ell_2$ in $L^p([0,1])$
Let $(g_n)_{n\geq 1}$ be a sequence of i.i.d. complex Gaussian random variables on $[0,1].$ Then it is easy to see that the map $j:\ell_2\to L^p([0,1])$ defined as $je_n=[E(g_n^p)]^{\frac{1}{p}}g_n,n\...
- 455
6
votes
1
answer
240
views
The approximation property for some spaces of holomorphic functions
I am reading a circle of papers which use arguments based on Fredholm determinants of nuclear operators to compute numerical quantities associated to real-analytic and holomorphic dynamical systems. ...
2
votes
1
answer
140
views
An inequality about embedding of cube into metric spaces
A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An edge of a cube is a pair of points $\{\psi(\epsilon_1),\psi(\epsilon_2)\}$ in $X$ such that $\epsilon_1$ and $\epsilon_2 $ differ in ...
4
votes
1
answer
157
views
Norm of "tensoring" with the identity
Consider a Banach space $E$ and a discrete set $X$. For an operator $T$ on $\ell^2(X)$ I can consider and induced operator $T'$ on the Bochner-Lebesgue space $\ell^2(X;E)$ of $E$-valued square-...
- 18.7k
12
votes
1
answer
575
views
Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?
Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...
- 538
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?
- 4,713
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
5
votes
1
answer
669
views
Compact operators on $\ell^1$
Let $T$ be a compact symmetric operator on $\ell^2$ and $T\vert_{\ell^1}$ be bounded on $\ell^1$. Are there any non-trivial conditions that $T\vert_{\ell^1}$ is compact as well (for example would $T$ ...
- 18.7k
5
votes
1
answer
462
views
Extending compact operators into $c_0$
Lindenstrauss has the following paper: http://www.ams.org/journals/bull/1962-68-05/S0002-9904-1962-10787-3/S0002-9904-1962-10787-3.pdf
I would like to see the proof for the following theorem (from ...
- 331
0
votes
1
answer
233
views
Is $(\ell^1(\mathbb N_0),\sigma(\ell^1,\ell^\infty))$ not quasi-complete?
In Jarchow's Locally Convex Spaces this not being quasi-complete is asserted on page 206 referring to Corollary 11.4.4 on page 228 saying that a Banach space is reflexive if and only if its closed ...
- 53.3k
3
votes
1
answer
151
views
The weakest condition guarantees some Separation-type of convex sets in Banach spaces
Classical Hahn-Banach Separation theorem plays a vital role in many branches of Analysis, Like functional Analysis, Convex Analysis, Variational Analyis, Theory of ODEs, optimal control and ...
- 4,786
5
votes
2
answers
216
views
On the coincidence (or non-coincidence) of two norms defined on the quotient of a given Hilbert $ C^{\ast} $-module by a certain linear subspace
Let $ A $ be a $ C^{\ast} $-algebra, $ I $ a closed two-sided ideal of $ A $, and $ \mathcal{E} $ a Hilbert $ A $-module. Let
$$
\mathcal{E}_{I}
\stackrel{\text{df}}{=}
\{ x \in \mathcal{E} \mid \...
- 830
8
votes
1
answer
434
views
Self-dual finite-dimensional complex normed spaces
Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space?
Remarks: There are easy counterexamples in the real case, and in ...
- 4,653
1
vote
0
answers
182
views
The real method of interpolation and operator ideals
Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...
2
votes
1
answer
233
views
complemented $\ell_p$ subspaces in $\ell_p$ sums of spaces
Note: By "subspace" I always mean an infinite-dimensional closed subspace.
Notation.
Let us write
$$\oplus_p\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{\ell_p}\;\;\;\text{ and }\;\;\;\...
- 11.3k
3
votes
0
answers
125
views
Commutative discrete cyclic operator groups on topological vector spaces
Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
- 63.9k
5
votes
0
answers
150
views
On the relation between Lipschitz free-spaces
Let $X$ be a pointed metric space, with base point 0. The space of Lipschitz function which preserves the base point,
$Lip_0(X)=\{f:X\to\mathbb{R} : f(0)=0\}$ consider with the norm $\|f(x)\|=\sup_{x\...
- 71