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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\...
Lviv Scottish Book's user avatar
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 $$ \limsup \|x_n - y\| < \limsup \|x_n\| ?$$ ...
Willie Wong's user avatar
  • 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 ...
user469053's user avatar
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 ...
Yaddle's user avatar
  • 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 $\|.\...
A beginner mathmatician's user avatar
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, ...
Dongyang Chen's user avatar
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 ...
Onur Oktay's user avatar
  • 2,605
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 ...
Neslihan's user avatar
  • 495
0 votes
2 answers
972 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 $...
Anupam's user avatar
  • 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 $\sigma$...
Ivan Feshchenko's user avatar
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 ...
Black's user avatar
  • 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 ...
Peter's user avatar
  • 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 $\ell_2$?
M.González's user avatar
  • 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?
Anupam's user avatar
  • 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 ...
Markus's user avatar
  • 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 ...
Blazej's user avatar
  • 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., $\lVert x_i\rVert_E=1$ ...
user92646's user avatar
  • 617
9 votes
1 answer
608 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 ...
Jan Bohr's user avatar
  • 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 $\...
Alessandro Codenotti's user avatar
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. ...
Matthias Ludewig's user avatar
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 ...
TheGeekGreek's user avatar
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 ...
user124297's user avatar
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)\...
Carlos_Petterson's user avatar
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....
S Argyros's user avatar
  • 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 ...
ABIM's user avatar
  • 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-...
Shridhar's user avatar
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 $(x_{n})_{n=1}^\infty$. ...
Dongyang Chen's user avatar
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?
Onur Oktay's user avatar
  • 2,605
5 votes
1 answer
952 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 ...
Guy Fsone's user avatar
  • 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 ...
M.González's user avatar
  • 4,461
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 ...
Whiteboard's user avatar
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 ...
Valerio Capraro's user avatar
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 ...
volond's user avatar
  • 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 ...
Sibyl Osullivan's user avatar
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\...
ABB's user avatar
  • 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 $D$ (...
Nicolò's user avatar
  • 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 ...
S Argyros's user avatar
  • 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 $...
jrranalyst's user avatar
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\...
Westlife's user avatar
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, $\...
e.lipnowski's user avatar
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). ...
ABIM's user avatar
  • 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 ...
Dongyang Chen's user avatar
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\|<...
Markus's user avatar
  • 1,361
4 votes
1 answer
396 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 ...
Robert Hermann's user avatar
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 ...
M.González's user avatar
  • 4,461
1 vote
0 answers
748 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 ...
HardyHulley's user avatar
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 ...
Anupam's user avatar
  • 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(...
Severin Schraven's user avatar
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 ...
T. Huynh's user avatar
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 ...
Overflowian's user avatar
  • 2,533

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