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2 answers
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Unbounded sequences in Banach spaces

Let $X$ be a Banach space and let $T$ be a bounded operator acting on $X$. Suppose for each linearly independent unbounded sequence $(x_n)$ in $E$, the sequence $(Tx_n)$ is unbounded. Must $T$ be ...
Olaf Kummers's user avatar
4 votes
1 answer
521 views

Basic sequences in $\ell_p$

Let $p\in [1,\infty)\setminus\{2\}$. Suppose $(e_n)$ is a basic sequence in $\ell_p$ (or $L_p$) equivalent to the basis of $\ell_p$ ($L_p$). Is there a subsequence $(e_{n_k})$ such that $[e_{n_k}]$ is ...
Olaf Kummers's user avatar
3 votes
2 answers
340 views

Perturbing upper-semi Fredholm operators

Let $T\colon X\to X$ be an upper-semi Fredholm operator acting on a $B$-space $X$ (the range of $T$ is closed and kernel is finite-dimensional) with complemented range. Suppose $S\colon X\to X$ is ...
Olaf Kummers's user avatar
2 votes
1 answer
373 views

Is it true that $c_0(X)^* = \ell_1(X^*)$ ?

I'm trying to prove this that but I can't . Any help/reference ?
Rafael's user avatar
  • 151
7 votes
2 answers
484 views

Extension of weakly compact operators from $\ell_1$ into $c_0$

Is every weakly compact operator from $\ell_1$ into $c_0$ extendible to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
Joaquin M. Gutierrez's user avatar
7 votes
1 answer
1k views

weak*-closed subspaces

Recall that a closed subspace $Y$ of a Banach space $X$ is weakly complemented if the set $$Y^{\bot}:= \{ f\in X^*| f(y) = 0 \forall y\in Y\}$$ is a complemented subspace of $ X^*$. For example, $c_0$ ...
Denis Poulin's user avatar
8 votes
2 answers
1k views

When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?

I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive ...
Miek Messerschmidt's user avatar
6 votes
2 answers
749 views

Transpose of unbounded operators between Banach spaces.

Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator $L' : \operatorname{...
shuhalo's user avatar
  • 5,327
14 votes
2 answers
6k views

Are weak and strong convergence of sequences not equivalent?

For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim_{i\to\...
TaQ's user avatar
  • 3,584
3 votes
1 answer
653 views

Converse of the taylor's theorem in Banach Spaces

I would like to known if the following converse of the taylor's theorem is true: Let $E$, $F$ Banach spaces, and $f:E\rightarrow F$ continuous. Suppose there are $k$ continuous functions $T_i: E \...
Ferraiol's user avatar
  • 121
2 votes
2 answers
816 views

Principle of Local Reflexivity

I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from 1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton) 2) ...
Rafael's user avatar
  • 151
2 votes
1 answer
414 views

Uniqueness of dimension in Banach spaces

Let $\;\;\; \big\langle V,+,\cdot, \;\; ||\cdot|| \;\; \big\rangle \;\;\;$ be a Banach space over the field $\mathbb{K}$, which is either $\mathbb{R}$ or $\mathbb{C}$. Suppose there exists a subset $...
user avatar
5 votes
1 answer
3k views

Weak convergence implying norm convergence

A surprising (to me) consequence of Hahn-Banach is that when a sequence converges weakly then there is another sequence made of (finite) convex combination which converges in norm (to the same element)...
ARG's user avatar
  • 4,432
11 votes
5 answers
5k views

A criterion for the sum of two closed sets to be closed ?

Let $V$ and $I$ be two closed subsets of a Banach space $A$. The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$. I would like to know whether $I+V$ ...
Fabien Besnard's user avatar
1 vote
0 answers
393 views

Unambiguous "weak" vector valued $L^{+\infty}$ spaces?

For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...
TaQ's user avatar
  • 3,584
1 vote
0 answers
149 views

Banach spaces with simple best approximate solutions

Let $\langle V,||.||\rangle$ be a Banach space such that: $\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$ $\;\;$ that ...
user avatar
15 votes
1 answer
1k views

Intersection of complemented subspaces of a Banach space

The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here. Question. Let $X$ be a Banach ...
Pietro Majer's user avatar
  • 60.5k
4 votes
1 answer
287 views

Second conjugate operators to operators on $c_0$

I posted my question at MS but unfortunately it is still without a response, so let me ask it here. We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,...
BSalkas's user avatar
  • 51
6 votes
0 answers
484 views

Square and cube?

Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...
Sellapan Nathan's user avatar
4 votes
1 answer
254 views

M-bases for $C(K)$-spaces, $K$ -scattered

Recall that a biorthogonal system $\{(x_i, x^*_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x_i\}_{i\in I}$ is linearly dense in $E$ and $\{x_i^*\}_{i\in I}$ separates points. Let me ...
Tomasz Kania's user avatar
  • 11.3k
7 votes
1 answer
682 views

$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$

Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum $A=\sum \mathcal{K}(\mathcal{H})$ of countably many ...
Habujew's user avatar
  • 113
17 votes
4 answers
2k views

Banach-Mazur applied to a Hilbert space

The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm. ...
Laurent Berger's user avatar
4 votes
1 answer
1k views

Projection exists ⇒ Uniformly convex?

I know that: Let X be a uniformly convex Banach space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x \right\...
Thomas Kuhn's user avatar
18 votes
1 answer
564 views

Is the space of Hankel operators complemented in B(H)?

Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$. Let ...
Yemon Choi's user avatar
  • 25.8k
15 votes
3 answers
8k views

What is an isomorphism of Banach spaces?

The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
Andrew Stacey's user avatar
43 votes
1 answer
5k views

Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?

Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
Lost's user avatar
  • 559
3 votes
2 answers
461 views

Complemented subspaces isomorphic to $c_0$ in $\mathcal{B}(E)$ [closed]

It is well known that neither 1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$ nor 2) $c_0$ is a quotient of $\mathcal{B}(H)$ for a Hilbert space $H$. Can we replace $H$ above ...
PhotonicCrystal's user avatar
0 votes
2 answers
225 views

Codimension of $J(\omega_1)$ in its bidual

I am reading the paper G. A. Edgar, A long James space, in: Measure Theory, Oberwolfach 1979, Lectures Notes in Math. 794, Springer-Verlag (1980) pp. 31-37. and I am pretty confused by the remarks ...
Briannon's user avatar
1 vote
1 answer
293 views

Basic sequences

Nowadays, we know that there exist Banach spaces without unconditional basic sequences. Do we know if something a bit milder holds? Namely, is that true each non-reflexive Banach space contains a ...
Sellapan Nathan's user avatar
12 votes
2 answers
547 views

Balls in spaces of operators

I am interested in some geometrical aspects of spaces $L(E)$, of bounded operators on a given Banach space $E$. I am unable to estimate if my problem deserves to be asked at MO, but let me try. Is ...
Sellapan Nathan'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
34 votes
1 answer
3k views

tr(ab)=tr(ba), part 2.

This is a Banach space version of Andre Henriques' question Trace Question for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
Bill Johnson's user avatar
  • 31.5k
6 votes
2 answers
2k views

How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space. Thanks in advance for any helpful answers.
q.g's user avatar
  • 71
0 votes
1 answer
498 views

Quotient of \ell_1 by space of finite sequences

The following question came up during a reading of Rudin's functional analysis. I have not been able to find any information through searching online, but I apologise if the answer is obvious, or the ...
Ivan's user avatar
  • 11
6 votes
0 answers
3k views

Projective and injective tensor product

It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space $(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$. If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
2 votes
1 answer
323 views

Recovering Schauder decompositions

The problem of Schauder decomposition of a given Banach space seems to play an important role in the geometry of Banach spaces, especially when one is interested in finite dimensional Schauder ...
TMK's user avatar
  • 23
4 votes
2 answers
484 views

When is a metric space isometrically embeddable into some Banach space?

EDIT Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something ...
Suvrit's user avatar
  • 28.6k
7 votes
3 answers
814 views

Preduals of B(E)

For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators. If $E$ is a Banach space then is it known whether $B(E)$ is always a ...
Ollie's user avatar
  • 1,411
6 votes
3 answers
2k views

Space of compact operators

I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...
Tomasz Kania's user avatar
  • 11.3k
5 votes
1 answer
578 views

Infimum over all vector-valued L^2 spaces

Suppose I have a Banach space $E$ (which may be finite dimensional if you wish), a Hilbert space $H$ and a tensor $\tau \in H\otimes E$ in the algebraic tensor product. There are lots of ways to ...
Matthew Daws's user avatar
  • 18.7k
1 vote
0 answers
369 views

Infinite internal direct sums of subspaces

Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a ...
Wiktor Jaszak's user avatar
44 votes
1 answer
4k views

Example of a compact set that isn't the spectrum of an operator

This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity: Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
Theo Buehler's user avatar
  • 5,743
1 vote
1 answer
312 views

Invertibility of frame/sampling operator on Bargmann-Fock spaces

Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
Joshua Isralowitz's user avatar
7 votes
2 answers
657 views

Subspaces isomorphic to $C[0, \omega_1]$

Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the ...
Tomasz Kania's user avatar
  • 11.3k
7 votes
1 answer
423 views

Best constant in comparison between Rademacher and gaussian averages?

Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables. What is the best ...
BigBill's user avatar
  • 1,222
0 votes
1 answer
365 views

Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
TaQ's user avatar
  • 3,584
0 votes
2 answers
796 views

Extending Continuous Sublinear maps on dense subsets of a Banach space

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying: (1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||. Please prove ...
Jeffrey's user avatar
  • 11
6 votes
1 answer
404 views

Unique preduals up to (nonisometric) isomorphism?

It's well known that there are Banach spaces which has a unique isometric predual-- for example, any von Neumann algebra. As other questions on here (for example, Isomorphisms of Banach Spaces ) ...
Matthew Daws's user avatar
  • 18.7k
10 votes
2 answers
881 views

volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product. Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
BigBill's user avatar
  • 1,222
5 votes
1 answer
794 views

Can the Sobolev norm of order 1/2 detect "jumps"?

We are given a function $f: \mathbb R^d \to \mathbb R$. For simplicity we can assume that $f$ is smooth and compactly supported. Is the Sobolev norm of order $\frac{1}{2}$ strong enough to prove an ...
Martins Bruveris's user avatar