The banach-spaces tag has no wiki summary.

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### 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

**1**answer

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 ...

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**1**answer

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 ...

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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 ...

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votes

**1**answer

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 ...

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**2**answers

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). ...

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**1**answer

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 ...

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**0**answers

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 $X\oplus X$. ...

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votes

**1**answer

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 ...

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**1**answer

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 ...

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**4**answers

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.
...

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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 ...

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votes

**1**answer

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 ...

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**2**answers

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 ...

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votes

**1**answer

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 ...

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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 ...

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**1**answer

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 ...

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**2**answers

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 ...

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votes

**1**answer

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 ...

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**2**answers

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 ...

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vote

**1**answer

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 ...

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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 ...

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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 ...

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**1**answer

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}
...

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**1**answer

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 $ab$ ...

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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.

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**1**answer

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 ...

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**1**answer

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 ...

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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 < ...

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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) ...

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**1**answer

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 ...

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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 ...

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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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 $M_n$, ...

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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 ...

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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 ...

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**1**answer

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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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vote

**1**answer

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} ...

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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 ...

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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 ...

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**1**answer

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 ...

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**1**answer

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 ...