All Questions
1,222 questions
8
votes
1
answer
523
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
5
votes
1
answer
506
views
Weak compactness of order intervals in $L^1$
Let $(\Omega,\mu)$ be a measure space, say $\sigma$-finite for the sake of simplicity, and let $L^1 := L^1(\Omega,\mu)$ denote the real-valued $L^1$-space over $(\Omega,\mu)$.
For all $f,h \in L^1$ ...
- 12.6k
3
votes
1
answer
269
views
What is a standard name for this kind of unconditional bases in Banach spaces?
I am looking for a standard name (if it exists) for the following property of a Schauder basis $(e_i)_{i=1}^\infty$ in a Banach space $X$:
$$\|\sum_{i\in F}x_ie_i\|\le\|x\|$$for any $x=\sum_{i=1}^\...
- 41.9k
0
votes
1
answer
328
views
Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407
1
vote
1
answer
228
views
Which norms on vectors can be consistently decomposed?
I need to know which permutation-invariant norms can be consistently decomposed in the sense that for any vector $v = (a,b,c)$ we have that
$$\|(a,b,c)\| = \|(\|(a,b)\|,c)\|.$$
More precisely, let $v ...
- 702
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
3
votes
1
answer
233
views
Extending linear isometries from subspaces of $\ell_p^n$
Take $p\in (1,\infty)\setminus \{2\}$. Let $X$ be a subspace of $\ell_p^n$ and let $U\colon X\to \ell_p^m$ ($m\geqslant n$) be a linear isometry. Is it possible to extend $U$ to a (non-surjective) ...
- 331
4
votes
2
answers
807
views
Completion of $\mathcal{S}(\mathbb{R})$ for a given norm
Assume that $\lVert \cdot \rVert$ is a norm on the space of rapidly decaying functions $\mathcal{S}(\mathbb{R})$. Under which conditions on the norm can we say that the completion $\mathcal{X}$ for ...
- 2,306
5
votes
0
answers
345
views
Weak to weak$^*$ continuity of the duality mapping
Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
- 799
5
votes
2
answers
263
views
How can we show that $E\:\hat\otimes_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\otimes_\pi\:E'\right)'$?
Let
$E$ be a $\mathbb R$-Banach space
$E\:\hat\otimes_\pi\:E$ denote the projective tensor product
How can we show that $E\:\hat\otimes_\pi\:E$ is isomorphic to a subspace of $\left(E'\:\hat\...
- 167
4
votes
1
answer
193
views
A bound on the square distance of a random walk on undirected graph
Fact:
Let $G$ be an $n$-vertex undirected graph and $(X_s)_{s\in \mathbb N}$ a stationary random walk on $G$. Then for every $s\in \mathbb{N}$,
$ \mathbb{E}[d_G(X_s,X_0)^2] \le C s \log n $, for some ...
- 401
9
votes
2
answers
338
views
Does $End(V)$ remember $V$, where $V$ is a locally convex space?
Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a ...
- 43.2k
0
votes
1
answer
221
views
A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded
We know every weakly open subset of an infinite-dimensional Banach vector space X is unbounded.
Now, Read's space $R$ (an infinite-dimensional Banach space) has the property:
there is $ρ >0$ such ...
- 93
3
votes
0
answers
422
views
Isometries between subspaces of finite-dimensional vector spaces
I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.
Taking $n \le m$, one ...
- 31
3
votes
0
answers
89
views
Discrete Lions Peetre interpolation
In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\...
user114263
2
votes
3
answers
256
views
Compact restrictions of the inclusion of $J:L_\infty(0,1)\to L_1(0,1)$
Given Banach spaces $X$, $Y$ and a bounded operator $T:X\to Y$ with non-closed range, a perturbation argument shows that there exists an infinite-dimensional closed subspace $M$ of $X$ such that the ...
- 4,461
4
votes
1
answer
956
views
Compact embedding for Sobolev space involving time
Let $d \in \mathbb{N}$ and $\Omega$ be a bounded domain of $\mathbb{R}^d$.
Consider $m,n,p,q \in \mathbb{N}$ and $T>0$.
Is the space $W^{m,p}([0,T],W^{n,q}(\Omega))$ compactly embedded in any ...
- 143
0
votes
0
answers
65
views
Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at ...
- 623
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
277
views
In Banach spaces is $X \cap Y = Z \Rightarrow \overline{{span} X} \cap \overline{{span} Y} = \overline{{span} Z}$
Let $V$ be a separable infinite dimensional Banach space over $\mathbb{C}$
Let $B \subset V$ be a subset of $V$ such that:
1) $B$ is linearly independent and closed
2) $\overline{\operatorname{span}...
- 433
0
votes
0
answers
57
views
A question on order unbounded sequences in Banach lattices
Let $E$ be a Banach lattice. It is well-known that every norm convergent sequence in $E$ admits an order convergent subsequence and hence admits an order bounded subsequence. But it seems that a norm ...
- 3,343
1
vote
1
answer
324
views
Question about a characterization of Grothendieck spaces
I do not believe the argument below is correct, but I am having quite a bit of trouble finding where I went wrong with this, so perhaps someone with more expertise in this area can push me in the ...
- 113
1
vote
0
answers
127
views
A point in Ion Suciu's paper on semigroups of isometric operators
My question is concerned a point in this 1968 paper by Ion Suciu which is given in Theorem 2. In the last paragraph of page 104, it is claimed that $N$ (given in the formula 2.5) is a wandering ...
- 4,058
5
votes
1
answer
256
views
Norming subspaces of duals of quotient spaces
In Davis, William J., and William B. Johnson. "Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces." Israel Journal of Mathematics 14.4 (1973): 353-367., the authors discuss the ...
- 151
6
votes
1
answer
238
views
Extending a weak*-converging sequence onto a superspace
Let $X$ be a real Banach space and $Y\subset X$ be a (closed) subspace of $X$. Assume that a sequence $y_n^*\in S_{Y^*}$ weak*-converges to some $y^*\in S_{Y*}$. (Here $S_{Y^*}$ stands for the dual ...
4
votes
2
answers
258
views
Duals of ideals of operators between Banach spaces
Given an operator ideal $\mathfrak{I}$, $\mathfrak{I}^\text{dual}$ is the class of all operators $A:X\to Y$ between Banach spaces $X$ and $Y$ such that $A^*\in \mathfrak{I}$. Given an operator ideal $\...
user114263
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 $f : [0,1] \to B$ ...
- 433
1
vote
1
answer
232
views
A double sequence in a Banach space
Let $V$ be a infinite dimensional Banach space over $\mathbb{C}$
Let $\{a_{m,n} \cdot v_{m,n}\}_{m,n \in \mathbb{N}}$ be a double sequence with $a_{m,n} \in \mathbb{C}$ and $v_{m,n} \in V$ such that:
...
- 433
1
vote
1
answer
394
views
Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$ and $\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalent norms?
Is it true that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms?
This results is pretty easy and straightforward for $...
- 1,101
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 ...
- 433
5
votes
1
answer
951
views
Comparison of two versions of fractional Sobolev spaces: do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?
There are two versions of fractional Sobolev spaces.
Definition 1: (Via Gagliardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and let $\Omega\subseteq \mathbb{R}^n$ be an open set. The ...
- 1,101
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, ...
- 51
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
813
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:
$$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$
...
- 4,058
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 ...
- 538
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
$$B(X)\to [0,\infty): T\to |\...
- 4,058
4
votes
1
answer
104
views
Two locally convex topologies on $B(X)$.
Let $X$ be a non-reflexive Banach space. It is supposed to compare two locally convex topologies on $B(X)$:
Let $w$ be the topology on $B(X)$ implemented by all seminorms given by
$$B(X)\to [0,\...
- 4,058
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
$$
\rho:\ell_p^n\to\ell_\infty(S, \ell_1^n),x\mapsto(f\cdot x)_{f\...
- 1,697
11
votes
1
answer
953
views
Separable bidual but nonseparable third dual
Does there exist a Banach space $X$ such that $X^{**}$ is separable but $X^{***}$ is non-separable?
More generally, for every natural $n$ can someone construct an example of Banach space $X$ such ...
- 521
1
vote
1
answer
790
views
$\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space
Let $X$ be a Banach space, and let $X'\subset X$ - its subspace. Then the following propositions are true:
$X'$ is closed, $X/X' \cong \ell_1 \Rightarrow X'$ is complementary;
$X' \cong \ell_\infty ...
- 51
5
votes
2
answers
853
views
Covering compactness in the weak sequential topology
Let $X$ be a real Banach space. Apart from the norm topology, we can consider the following weak topologies on $X$:
the weak toplogy, defined as the initial topology with respect to $X^*$. In other ...
- 171
1
vote
1
answer
96
views
About representations of some elements in $\mathcal A(\ell^p)$
For Banach spaces $E$ and $F$ we denote the approximate operators by $\mathcal A(E,F)$ and projective tensor product by $\hat\otimes$.
Consider the natural map
$$\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 ...
- 3,343
7
votes
3
answers
753
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 ...
- 423
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
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
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 ...
- 5,529
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