All Questions
1,222 questions
11
votes
1
answer
227
views
Complemented subspaces of $C(\beta\mathbb N\times \beta\mathbb N)$
Problem. Is there any complemented subspace in the Banach space $C(\beta\mathbb N\times\beta\mathbb N)$, not isomorphic to $c_0$, $c_0\oplus C(\beta\mathbb N)$, $C(\beta\mathbb N)$, $c_0(C(\beta\...
- 4,535
12
votes
1
answer
467
views
Subtracting the weak limit reduces the norm in the limit
Question
Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that
$$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$
...
- 39.1k
1
vote
0
answers
74
views
Multiple steps of the Gorelik principle
The following result is one of several non-linear Banach space theory results known as The Gorelik Principle. I am stating it here in a weaker form than what is in the literature, but the statement ...
- 81
9
votes
0
answers
540
views
Why is spectral theory developed for $\mathbb C$
Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
- 381
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
3
votes
1
answer
452
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
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 495
4
votes
1
answer
254
views
Separable subalgebras of non-separable reflexive Banach algebras
Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [Lindenstrauss,1966]. It is straightforward to show that every ...
- 2,605
0
votes
2
answers
974
views
Example of a linear operator whose graph is not closed
I want an example of a linear operator $T:X\to Y$ such that graph of $T$ is not closed.
My thoughts: $T$ must be unbounded. Again by closed graph theorem any unbounded linear map from a Banach space $...
- 585
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
5
votes
1
answer
1k
views
Wildly discontinuous linear functionals
Let $X$ be a Banach space, $H\subseteq X$ be a dense hyperplane, and $f$ be a continuous linear functional defined on $H$. Then $f$ is uniformly continuous and hence it admits a unique continuous ...
- 483
7
votes
1
answer
1k
views
Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?
Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
- 141
7
votes
1
answer
195
views
Self-dual Orlicz sequence spaces
Suppose that $\ell_\phi$ is a reflexive Orlicz sequence space such that its dual space $\ell_\phi^*$ is isomorphic to $\ell_\phi$.
Is $\ell_\phi$ isomorphic to $\ell_2$?
- 4,461
5
votes
2
answers
861
views
Injective continuous operators between Banach spaces
Suppose $X$ and $Y$ are two infinite dimensional Banach spaces. What can we say about the set of all injective continuous linear operators between $X$ and $Y$? Is it always nonempty?
- 585
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
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
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
9
votes
1
answer
610
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
5
votes
1
answer
247
views
How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
- 3,011
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
4
votes
1
answer
566
views
Fréchet vs. Carathéodory differentiability on Banach spaces
It is well-known that in the finite-dimensional case one can use the notion of Fréchet differentiability and Carathéodory differentiability interchangeably. See for example the 194 AMM article Frechet ...
- 548
2
votes
0
answers
76
views
Fractional integration in Orlicz spaces
I am reading the paper "Fractional integration in Orlicz spaces" by R. Sharpley.
And I would like to understand one question:
Let $A,B, C$ are Young's functions. The spaces $L_A, L_B$ are ...
- 343
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
8
votes
0
answers
246
views
A question related to the separable quotient problem
I have the following question related to the previous posts Hereditarily indecomposable Banach spaces and Separable Quotient problem and Weak star separable and separable quotient problem
Question....
- 986
5
votes
3
answers
675
views
$L^{\infty}$ as colimit
I recently read a result (in Jarchow's book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following.
Let $\mu$ be a ...
- 5,405
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
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
4
votes
1
answer
119
views
Is the "hereditarily indecomposable" property separably determined?
Is it true that a Banach space $X$ is hereditarily indecomposable if every separable closed subspace of $X$ is hereditarily indecomposable?
- 2,605
5
votes
1
answer
953
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
2
answers
177
views
Subprojective Orlicz sequence spaces
A Banach space $X$ is subprojective if every infinite dimensional closed subspace $Y$ of $X$ contains an infinite dimensional subspace $Z$ which is complemented in $X$.
I am interested in conditions ...
- 4,461
28
votes
3
answers
4k
views
A separable Banach space and a non-separable Banach space having the same dual space?
I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is ...
- 6,034
5
votes
1
answer
855
views
$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
- 51
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
6
votes
1
answer
167
views
Extension Operator for $W^{1,\infty}(U,X)$
I am reading through some lectures on Sobolev spaces and the vector-valued (or Banach space valued) version of them. At this moment I am very interested in extension operators for the vector-valued ...
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
19
votes
6
answers
8k
views
Unbounded operator bounded in a dense subset
Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
- 783
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
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
3
votes
1
answer
951
views
Specific criterion for the sum of two closed sets to be closed
Let $Y$ and $Z$ be two closed subspaces of a Banach space $X$ with $Y\cap Z=\{0\}$.
I know that $Y+Z$ is a closed subspace of $X$ $\iff \exists \alpha > 0:\quad \lVert y\rVert \le \alpha\lVert y+z\...
- 33
9
votes
1
answer
333
views
Closedness of linear image of positive L1 functions
Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
- 537
1
vote
0
answers
97
views
Are Hölder functions between Banach spaces residual in the compact-open topology?
Let $X$ and $Y$ be Banach spaces and let $C(X,Y)$ be the set of continuous functions from $X$ to $Y$ equipped with the topology of uniform convergence on compact sets (i.e. the compact-open topology). ...
- 5,405
6
votes
2
answers
509
views
A question on Grothendieck space
A Banach space $X$ is said to be Grothendieck if the weak and the weak* convergence of sequences in $X^{*}$ coincide. I have the following two questions.
Question 1. A Banach space $X$ is Grothendieck ...
- 3,343
3
votes
1
answer
157
views
Operator in the commutant which is small on a given vector
Suppose $x$ is a non-zero vector in a Banach space, and $T$ is a fixed operator. Is the following true:
For any $\varepsilon, \delta$, there exists $S$ in the commutant of $T$ such that $1\leq\|S\|<...
- 1,361
4
votes
1
answer
398
views
Closedness of the image of the unit ball
Let $X$ be a Banach space and let $P$ be a bounded, linear projection on $X$. Is $P[B_X]$ closed in $X$? Here $B_X$ is the closed unit ball of $X$.
This is trivial if $X$ is reflexive, but otherwise ...
1
vote
1
answer
330
views
Duality $(M/N)^*\equiv N^\perp/M^\perp$ for closed subspaces $N\subset M$ of a Banach space
Let $M$ be a closed subspace of a Banach space $X$. Then we can identify $(X/M)^*$ with $M^\perp$ and $M^*$ with $X^*/M^\perp$.
Indeed, if $Q^*:X\to X/M$ is the quotient map, then $Q^*:M^*\to X^*$ is ...
- 4,461
1
vote
0
answers
749
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
5
votes
1
answer
923
views
Existence of injective compact operators
We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$.
My query is for every Banach ...
- 585
6
votes
1
answer
323
views
Hartogs' theorem in Banach spaces
In complex analysis one learns Hartogs' theorem:
Let $U\subseteq \mathbb{C}^n$ open and $f: U \rightarrow \mathbb{C}$ a function. Then $f$ is analytic iff for all $1\leq i \leq n$
$$ z \mapsto f(...
- 1,045
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
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