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5
votes
1answer
379 views

Derandomizing random matrices

My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates ...
6
votes
1answer
392 views

Do real vectors attain matrix norms?

I apologize if the following question ends up being too elementary for this website; I asked it on math.SE a week ago and it remains unanswered. Let $A$ be an $n \times n$ matrix with real entries ...
0
votes
1answer
220 views

Second adjoint operators on non quasi-reflexive Banach spaces

I am interested in 'algebraic-density'-type properties of second adjoint operators in the algebra of bounded operator on a second dual of a Banach space. Incidentally, I have a problem with ...
1
vote
0answers
141 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 ...
12
votes
1answer
615 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 ...
1
vote
2answers
280 views

ED compact $K$ such that $C(K)$ is not a dual Banach space

Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...
4
votes
1answer
247 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 ...
4
votes
0answers
365 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 $X\oplus X$. ...
4
votes
1answer
155 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 ...
6
votes
1answer
357 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 ...
15
votes
4answers
909 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. ...
5
votes
2answers
468 views

Continuous choice of Hahn-Banach extensions

QUESTION RETRACTED - My original argument was fundamentally mistaken (mixing up lower and upper semi-continuity). Sorry (and thanks for the useful comments) I need, and (unless I am seriously ...
2
votes
1answer
584 views

Projection exists => Uniformly convex?

Hello, 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 ...
3
votes
2answers
297 views

homogenuity of $\ell^p$

I want to know the following: If $x_1, x_2, \cdots, x_n, y_1,y_2, \cdots, y_n \in \ell_p$ satisfies $\|x_i-x_j\|_p=\|y_i-y_j\|_p$ for any $i,j$, then does there exist isometry $F$ of $\ell_p$ which ...
13
votes
1answer
350 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 ...
10
votes
3answers
2k 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 ...
1
vote
1answer
691 views

A question about density character of Banach spaces. [closed]

Let $\langle M_i:i<\theta\rangle$ be an increasing chain of Banach spaces, where each $M_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M_i$ is $\mu$). Let ...
2
votes
2answers
350 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 ...
3
votes
1answer
767 views

Complemented subspaces of Banach spaces

It is known (Lindenstrauss, Tzafriri, On the complemented subspaces problem) that a real Banach space all of whose closed subspaces are complemented (i.e. have a closed supplement) is isomorphic (as a ...
0
votes
2answers
188 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 ...
1
vote
1answer
221 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 ...
9
votes
2answers
481 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 ...
13
votes
3answers
1k 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 ...
3
votes
1answer
328 views

injective tensor norms for real tensors

If $A$ is an element of $\mathbb{R}^n \otimes\mathbb{R}^n \otimes\mathbb{R}^n$, then define its injective tensor norm to be $$\|A\|_{\rm inj} := \max_{x,y,z\in \mathbb{R}^n, \|x\|=\|y\|=\|z\|=1} ...
27
votes
1answer
2k 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 $ab$ ...
3
votes
2answers
677 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.
1
vote
1answer
246 views

Projections in Sobolev spaces

I was browsing through the literature but I have not found anything related to my question: I am interested in decompositions of functions in Sobolev spaces $W^{k,p}(\Omega)$, where $\Omega$ is some ...
0
votes
1answer
408 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 ...
3
votes
0answers
1k 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 < ...
0
votes
0answers
338 views

Surjectively isometric normed spaces: Hamel vs (extended) Schauder dimension

Bonjour/bonsoir à toutes et à tous. This may really be a very basic question, but... Let $\mathbf{X} \equiv (X, \|\cdot\|_X)$ and $\mathbf{Y} \equiv (Y, \|\cdot\|_Y)$ be surjectively isometric (1) ...
2
votes
1answer
225 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 ...
4
votes
2answers
347 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 ...
3
votes
2answers
403 views

“Measuring” how far is one Banach space from being surjectively isometric to another

Bonjour/bonsoir à toutes et à tous. Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field). Question 1. What ...
15
votes
0answers
713 views

Decomposable Banach Spaces

An infinite dimensional Banach space $X$ is decomposable provided $X$ is the direct sum of two closed infinite dimensional subspaces; equivalently, if there is a bounded linear idempotent operator on ...
2
votes
1answer
365 views

Non-isometric Banach spaces [closed]

I am sorry if the question is easy but can one give me an example of a pair of Banach spaces, say $X$ and $Y$, $X$ isomorphic to $Y$ such that $X$ has no isometric copy of $Y$ neither $Y$ has ...
3
votes
2answers
359 views

Subspaces of duals

It is an easy undergraduate exercise to show that (finite) direct sums are preserved under dualisation. Thus, it is natural to ask if we the following holds: is it true that if $X$ is a subspace of ...
3
votes
1answer
230 views

Ultraproduct of n-dimensional Banach spaces and algebras

Hi, I am interested in the following question: Fix $n$. Let $M_n$ be matrix algebra over the field of complex numbers with normalized trace $tr_n$. Let $M_n^{\omega}$ be an ultrapover of $M_n$, ...
7
votes
2answers
1k views

Uniformly Convex spaces

My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill). I'm interested in constructing a uniformly convex Banach space with conditional structure ...
7
votes
3answers
600 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 ...
1
vote
1answer
361 views

Extensions of Banach spaces

I am looking for an answer to the following questions: Are there infinite-dimensional Banach spaces $X$ and $Y$ for which there are non-split extensions $0 \to X \to E_1 \to Y \to 0$ and $0 \to X ...
0
votes
0answers
202 views

Square variation norm and non-negative, non-increasing sequences

I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual ...
6
votes
2answers
939 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 ...
5
votes
1answer
434 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 ...
1
vote
0answers
267 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 ...
26
votes
0answers
2k 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 ...
1
vote
1answer
221 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} ...
5
votes
2answers
552 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 ...
6
votes
1answer
267 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 ...
0
votes
1answer
222 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 ...
3
votes
1answer
303 views

Completely bounded maps on Mn

The aim of this question is to collect nice maps on $M_n(\mathbb{C})$ with the following property: $\phi_n:M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})$ with $||\phi||=1$ and ...