All Questions
1,222 questions
3
votes
0
answers
81
views
Example of the bounded convolution operator when Sharpley's conditions does not hold
I am reading about Orlicz and Marcinkevich spaces, and wondering whether there is an example in which Sharpley's condition is not satisfied for a special bounded operator $T_k$ (see for reference ...
- 97
4
votes
1
answer
244
views
Boundedly complete bases
Let us recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup\limits_{n}\|\sum\limits_{i=1}^{n}a_{i}x_{i}\|<\infty$, ...
- 3,343
0
votes
0
answers
36
views
Regarding significance of spectral variation under algebraic operations
I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties.
The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach ...
- 27
1
vote
0
answers
85
views
Interpolation between projective and injective spaces
Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex ...
- 1,879
4
votes
1
answer
405
views
Why is it difficult to define a direct integral of Banach spaces or Banach algebras?
In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or ...
- 1,054
3
votes
0
answers
257
views
Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
1
vote
3
answers
580
views
Squeezing more convergence from the convergence in all $L^p$ spaces
Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
- 5,407
7
votes
1
answer
200
views
Projective tensor product of injective operators
I've seen claims that it is known that for a pair of bounded injective linear operators $T\colon X\to Y, S\colon W\to V$, their tensor product $T\otimes S\colon X \otimes_\pi W\to Y \otimes_\pi V$ ...
- 11.3k
1
vote
0
answers
111
views
Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
5
votes
0
answers
169
views
Is the Grassmannian of a Banach space an infinite dimensional manifold?
Grassmannian of complemented subspaces in a Banach space is a Banach manifold. This is explained for example in the thesis of Douady and is rather analogous to the finite-dimensional case.
I would ...
- 344
4
votes
1
answer
490
views
ODE in Banach space
Have I understood this correctly:
So originally we consider the following partial differential equation:
$$u_t= \frac{u_{xx}}{1+w}-\frac{1}{\epsilon}(1+w)u^3+\frac{wu}{\epsilon(1+w)} \text{ in } \...
- 41
4
votes
1
answer
281
views
Does property (V) imply the Grothendieck property for dual Banach spaces?
A Banach space $X$ has property (V) whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to ...
- 4,461
1
vote
1
answer
214
views
An explicit description for a certain type of infinite-dimensional homogeneous polynomials
This is a side question from Infinite-dimensional "algebraic varieties".
Denote by $X_p$ ($1 \le p \le \infty$) the Banach spaces of complex sequences with finite $p$-norm and limit $0$. ...
- 1,543
6
votes
1
answer
237
views
Radon-Nikodym property in Diestel & Uhl: a definition clarification
I posted the following question on MSE originally because, not being research-level, it seemed more appropriate for that site. However, there was no activity for it on MSE and I feel that it certainly ...
- 201
1
vote
1
answer
155
views
Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
- 1,054
1
vote
0
answers
747
views
Notation for the space of eventually-zero sequences
An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
- 153
2
votes
1
answer
259
views
Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?
It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
- 319
3
votes
0
answers
278
views
Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 67
4
votes
1
answer
212
views
$c_{0}$ has no boundedly complete basis
Recall that a basis $(x_{n})_{n}$ for a Banach space $X$ is called boundedly complete if for every scalar sequence $(a_{n})_{n}$ with $\sup_{n}\|\sum_{i=1}^{n}a_{i}x_{i}\|<\infty$, the series $\...
- 3,343
9
votes
0
answers
1k
views
Weak compactness in $\mathcal{F}(X)$
Let $(X,0)$ be a pointed metric space and let $\mathcal{F}(X)$ be the natural predual of ${\rm Lip}_0(X)$, the space of Lipschitz functions on $X$ that map $0$ to $0$; here $\mathcal{F}(X)$ is really ...
- 11.3k
1
vote
1
answer
452
views
Uniformly convex Banach spaces
Theorem. If $X$ and $Y$ are uniformly convex Banach spaces, then for $1<p<\infty$ the space
$$
X\oplus_pY=X\times Y,
\qquad
\Vert(x,y)\Vert:=(\Vert x\Vert_X^p+\Vert y\Vert_Y^p)^{1/p}
$$
is ...
- 28k
0
votes
0
answers
241
views
Can a non-reflexive space embed into a reflexive space?
My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
- 11
15
votes
2
answers
931
views
Distinguishing topologically weak topologies of Banach spaces
Are the weak topologies of $\ell_1$ and $L_1$ homeomorphic?
Strangely may it sound, the question seeks contrasts between norm and weak topologies of Banach spaces from the non-linear point of view. ...
- 11.3k
3
votes
2
answers
135
views
Unicellular compact operators
An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...
- 1,361
4
votes
2
answers
428
views
Are sequences in $\ell^1(\mathbb N_0)$ converging uniformly on convex weakly compact subsets of $c_0(\mathbb N_0)$ norm convergent?
I think the question as expressed in the title should be clear. I do not know whether there is a known "characterization" of the weakly compact convex sets in $c_0(\mathbb N_0)$ but testing ...
- 3,584
1
vote
0
answers
82
views
Extreme case of K-interpolation
Suppose $X_0$ and $X_1$ are Banach spaces living in a larger Banach space
$X$. The $K$-functional is defined for each $f\in X_0+X_1$ and $t>0$ as
$$K(f,t,X_0,X_1)=\inf\{\|f_0\|_{X_0}+t\|f_1\|_{X_1}:...
- 319
3
votes
0
answers
92
views
Asymptotic uniform convexity conditions for subsets of the $B_X$
The following question is relatively straightforward and almost looks like an exercise from a textbook but I have no idea how to handle it. The problem is related to spaces with asymptotically ...
- 1,073
3
votes
0
answers
61
views
Dual space of Carleman functions
Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which
$$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
- 141
3
votes
0
answers
108
views
$ f,g\in \mathrm{VMO} $ but $ f\cdot g\notin \mathrm{VMO} $
We say a function $ f\in L^1_{\mathrm{loc}}(\mathbb{R}) $ is in $\mathrm{BMO}(\mathbb{R})$ if
$$\|f\|_{\mathrm{BMO}}=\sup_{I}\frac{1}{|I|}\int\limits_I |f(y)-f_I|\, dy<\infty$$ for all intervals $I\...
- 1,219
1
vote
1
answer
116
views
Definition of $1$-spreading basis and spreading model
I recall two definitions from Banach space theory
Definition 1. Let $E$ be a Banach space, then a basis $(e_n)_{n\in\mathbb{N}}$ of $E$ is called $1$-spreading if $$\left\|\sum_{i=1}^k a_i e_{m_i}\...
- 2,286
2
votes
1
answer
136
views
Eigenvectors of the dual of positive irreducible operators
This question was previously posted on MSE.
Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with ...
- 333
5
votes
4
answers
362
views
Dual norm of a subspace of $\ell_\infty^3$
We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
- 1,879
2
votes
1
answer
151
views
Banach-Mazur distance between Schatten-$p$ classes
Let $M_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M_n$ define $\|A\|_p:=(Tr(A^*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\...
- 1,879
1
vote
0
answers
163
views
Infinite matrices from $\ell^p$ to $\ell^{p/(p-1)}$ that are compact operators
I wanted to ask if my proof (sketch) of the following statement is correct. Namely, let $p>1$ and define $q= \frac{p}{p-1}$ we are given an operator $K : \ell^{p} \rightarrow \ell^{q}$ defined as $...
- 11
5
votes
0
answers
280
views
Completeness of the space $L^p$ and the Axiom of Countable Choice
I am thinking about the proof that the usual $L^p$ spaces are complete.
So, let $(X,\mathcal{F},\mu)$ be a measure space and let
$p\in[1,+\infty)$.
Important: by a measure I mean a nonnegative $\sigma$...
- 935
1
vote
0
answers
292
views
Closure of finite rank operators on $L^p$
It well-known that, an operator $T:H\to H$ on a Hilbert space, is compact if and only if T is limit of finite rank operators.
Besides this, the results by Per Enflo 1973 shows that this results is ...
- 1,101
4
votes
0
answers
212
views
"Cyclic vector" of sequence of operators
I recently encountered the following somewhat random-looking problem in my research. At first I thought that should not be too hard, but now, the more I think about it, the more interesting it seems.
...
- 9,853
10
votes
1
answer
368
views
Group of isometries of Banach spaces a topological group?
Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$.
Q: Is $\mathrm{Iso}(X)$ a topological group ...
- 9,853
0
votes
1
answer
233
views
When does $C_b(X)$ admit a Schauder Basis?
Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x_n,\cdot)-d(x_0,\cdot)\...
- 195
1
vote
1
answer
123
views
Bayesian inverse problems on non-separable Banach spaces
I am now studying Bayesian inverse problems. In the note of Dashti and Stuart https://arxiv.org/abs/1302.6989, they mentioned that "... when considering a non-separable Banach space $B$, it is ...
- 41
2
votes
0
answers
134
views
Fourier type of asymptotic-$\ell_{2}$ Banach spaces
A Banach space $X$ is said to have Fourier type $p\in[1,2]$ if the Fourier transform $\hat{f}(s):=\int_{\mathbb{R}}e^{-ist}f(t)dt$ defines a bounded linear operator from $L_{p}(\mathbb{R},X)$ to $L_{p'...
- 201
3
votes
1
answer
451
views
Bases in $c_{0}$
$c_{0}$, the space of the scalar sequence that converges to $0$ endowed with the sup norm, has two well-known bases: the unit vector basis $(e_{n})_{n}$, where $e_{n}(k)=1$ if $k=n$ and $0$ otherwise, ...
- 3,343
1
vote
1
answer
232
views
An approximation property in a separable topological vector space
Let $X$ be a topological vector space.
Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
- 4,058
0
votes
0
answers
103
views
A question on the Haar basis for $L_{1}[0,1]$
Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. It is important to know the exact expression of the norm of $\|\sum_{i=1}^{n}a_{i}x_{i}\|$ for all $n$ and all scalars $a_{1},a_{2},\ldots,...
- 3,343
4
votes
0
answers
132
views
$L_1$-subspace of the predual of a von Neumann algebra
If $M$ is a type $II$ von Neumann algebra, then the predual has a complemented subspace isometric to $L_1(0,1)$. It follows from the existence of expectation. However, I don't know whether such a ...
- 617
1
vote
1
answer
130
views
Quantifications of boundedly complete bases
Let $(x_{n})_{n=1}^\infty$ be a bounded sequence in a Banach space $X$. We set
$$\textrm{ca}((x_{n})_{n=1}^\infty)=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$
Then $(x_{n})_{n=1}^\infty$ is norm-Cauchy ...
- 3,343
3
votes
1
answer
141
views
Quantifying shrinking bases
Let $X$ be a Banach space and let $(x_{n})_{n=1}^\infty$ be a (Schauder) basis for $X$. Let $(x^{*}_{n})_{n=1}^{\infty}$ be the biorthogonal functionals associated to the basis $(x_{n})_{n=1}^\infty$. ...
- 3,343
14
votes
2
answers
873
views
Which finite dimensional Banach spaces can be represented isometrically as spaces of bounded operators on a finite dimensional Hilbert space?
Background:
It is known that every Banach space $X$ can be embedded isometrically as a subspace in the space $C(K)$ of continuous functions on a compact Hausdorff space $K$. Indeed, one can take $K$ ...
- 550
4
votes
0
answers
184
views
Weak* HI Banach spaces
The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable. (See the previous posts Decomposable Banach Spaces, Hereditarily ...
- 986
2
votes
1
answer
144
views
On the symmetric basic sequence of a symmetric sequence space
Let $E$ be a separable Banach space with symmetric basis $\{e_i\}$ (it is also called a symmetric sequence space).
Let $\{x_i\}$ be a normalized disjoint sequence in $E$, i.e.,
$\lVert x_i\rVert_E=1$ ...
- 617