**12**

votes

**0**answers

230 views

### Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?

$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic.
Is this folklore, or is it credited to someone? ...

**0**

votes

**0**answers

53 views

### Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...

**1**

vote

**0**answers

102 views

### When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto ...

**4**

votes

**1**answer

237 views

### Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, ...

**2**

votes

**0**answers

52 views

### Two questions on the James $p$-space $J_{p}(1<p<\infty)$

Let $1<p<\infty$. The James $p$-space $J_{p}$ is the Banach space of all sequences of real numbers $(a_{i})_{i}\in c_{0}$ such that
...

**5**

votes

**0**answers

125 views

### Constructing Extreme Points in Reflexive Banach Spaces

A theorem of Lindenstrauss and Phelps states that if $X$ is a separable reflexive Banach space then the unit ball of $X$, $Ba(X)$, has uncountably many extreme points. The proof goes by contradiction ...

**1**

vote

**1**answer

54 views

### Various limits of the Christoffel Darboux Kernel

In a different thread, we stumbled upon the following question:
Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...

**7**

votes

**2**answers

350 views

### Attempted Banachification of a space

In a recent blog post, Terry Tao mentioned the question of how to tell if a Hausdorff topological vector space admits a finer topological structure which happens to be the topology of a Banach space ...

**4**

votes

**0**answers

85 views

### Good reference for noncommutative $L^p$ spaces

I'm looking for good references to learn about $L^p$ spaces associated with von Neumann algebras. I already know about Uffe Haagerup's paper "$L^p$-spaces associated with an arbitrary von Neumann ...

**4**

votes

**0**answers

71 views

### On the weakly sequential completeness of the dual of the James space $J$

Let me first introduce some definitions. Let $1\leq p\leq \infty$.
A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be weakly $p$-convergent to $x\in X$ if the sequence $(x_{n}-x)_{n}$ is ...

**8**

votes

**2**answers

301 views

### $l^1$ versus $l^2$

Is there an elementary proof of this Banach space fact?
If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite ...

**2**

votes

**0**answers

78 views

### Approximate constructive spectral theorem

Let $T: H \rightarrow H$ be a Hermitian operator acting on an $n-$dimensional Hilbert space space $H$ with the spectrum $\lambda_1, ... \lambda_m, m \leq n$. It follows classically that
$$ T = ...

**3**

votes

**1**answer

63 views

### Unconditionally $p$-converging operators on $L_{1}[0,1]$

Let $1\leq p<\infty$. We say that an operator $T:X\rightarrow Y$ is unconditionally $p$-converging if $T$ takes a weakly $p$-summable sequence to a norm null sequence.
Question: Is every ...

**3**

votes

**1**answer

215 views

### The dual space of $C[0,1]$

Since $C[0,1]^{*}$ is an abstract $L$-space, $C[0,1]^{*}$ is order isometric to $L_{1}(\mu)$ for some measure $\mu$.
My question is: the measure $\mu$ can be choosen to be a finite positive measure?
...

**12**

votes

**2**answers

868 views

### Do non-stable Banach spaces exist?

Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:
Is every infinite ...

**1**

vote

**0**answers

52 views

### Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces".
We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...

**2**

votes

**0**answers

58 views

### Maurey-Pisier Theorem for complex Banach spaces

A famous theorem of Maurey and Pisier, usually stated for real Banach spaces, says that $\ell_a$ and $\ell_b$ are finitely representable in $X$, where $a$ is the supremum of the $p$ such that $X$ has ...

**3**

votes

**1**answer

286 views

### Invariant probability on a unit ball of a Banach space

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.
Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) ...

**0**

votes

**1**answer

85 views

### Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$

Let $X$ be a Banach space with basis $(e_n)_{n=1}^\infty$, and suppose that $(x_i)_{i=1}^\infty$ is a normalized block basic sequence of $(e_n)_{n=1}^\infty$. In addition assume that ...

**4**

votes

**0**answers

114 views

### A Banach space with the BD property and without the weak Gelfand-Phillips property

A subset A of X is called Grothendieck if every operator T from X to $c_0$ maps A to a relatively weakly compact set.
A Banach space has the weak Gelfand-Phillips property (wGP) if every ...

**4**

votes

**1**answer

174 views

### Non-equivalence of admitting different types of bases in Banach spaces

Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis ...

**1**

vote

**3**answers

99 views

### Classification of subsymmetric basic sequences

I just encountered the following statement: any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell_1$.
It's the first time I realize that. I do see the ...

**0**

votes

**0**answers

40 views

### A question on measuring weak non-compactness in $L_{1}(\mu)$ spaces

Let me first fix some notations. Let $X=L_{1}(\mu)$, where $\mu$ is a finite positive measure. Let $A$ be a bounded subset of $X$. Set
$\omega(A)=\inf\{\widehat{d}(A,K):\emptyset\neq K\subset X$ is ...

**1**

vote

**2**answers

148 views

### Where can I find some articles and lecture notes in renorming theory in Banach spaces? [closed]

I am really into renorming theory in Banach spaces especially, renorming in non-reflexive Banach spaces such that they have nice property, for example they have fixed point property,locally uniformly ...

**0**

votes

**0**answers

53 views

### Existence of strictly convex Banach space without weak normal structure

This is a complementary question to the following at
Existence of normal structure in strictly convex Banach spaces
Does there exist a strictly convex Banach space without the weak normal structure ...

**4**

votes

**1**answer

144 views

### Is the topological dual of a Banach space weakly* closed in its algebraic dual?

The question is completely contained in the title :)
I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential ...

**11**

votes

**0**answers

258 views

### Kolmogorov width for cartesian products

For an operator $T:X\to Y$ between Banach spaces with unit balls $B_X$ and $B_Y$ the sequence of Kolmogorov widths is
$$
\delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...

**0**

votes

**1**answer

90 views

### Spectrum of compact operator between different Banach spaces

Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ is well-defined. (For example, we could have $X = L^2([0,1])$ and $Y = ...

**23**

votes

**0**answers

478 views

### Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $||x|| = 1$ and $||Ax|| = ||A||$. The ...

**1**

vote

**2**answers

234 views

### $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$

$c_0(C[0,1])$ is the $c_0$-direct sum of countably many $C[0,1]$.How to prove
$C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$.
Here,Banach-space isomorphism means a bounded invertible operator ...

**1**

vote

**0**answers

47 views

### weak compactness in $(\sum_{n=1}^{\infty}\oplus X_{n})_{p}(1<p<\infty)$

The following result is well-known:
Let $(X_{n})_{n}$ be a sequence of Banach spaces and let $X=(\sum_{n=1}^{\infty}\oplus X_{n})_{p}(1<p<\infty)$. Then a bounded subset $A$ of $X$ is ...

**11**

votes

**1**answer

178 views

### Weak*-closure of finite rank operators on dual space

Given a Banach space $X$, we consider the space $B(X^*)$ of bounded, linear operators on $X^*$ with the weak*-topology from its canonical predual $B(X^*)_*=X^*\hat{\otimes}X$. What is ...

**2**

votes

**0**answers

51 views

### A quantitative version of Pełczyński's property ($V^{*}$)

Let me first fix some notations. Let $A$ be a bounded subset of a Banach space $X$. Set $$wk_{X}(A)=\widehat{d}(\overline{A}^{w^{*}},X)=\sup\{d(x^{**},X):x^{**}\in \overline{A}^{w^{*}}\},$$ where ...

**4**

votes

**0**answers

81 views

### Weakenings of the Bounded Approximation Property

Let $X$ be a Banach space, and let $F(X)$ be the finite-rank operators on $X$.
Then $X$ has the $\lambda$-BAP if there exists a net $(S_i)_i$ in $F(X)$ with $\sup_i\|S_i\|\leq\lambda$ such that ...

**10**

votes

**1**answer

216 views

### Nonseparable Hilbert spaces

Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...

**1**

vote

**0**answers

107 views

### A quantity measuring weak compactness

Let us fix some notations. If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $d(A,B)=\inf\{\|a-b\|\mid a\in A,b\in B\},\widehat{d}(A,B)=\sup\{d(a,B)\mid a\in A\}.$ Let $A$ be a ...

**0**

votes

**1**answer

153 views

### Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$
Of course if $f\in H^s(\mathbb{R}^3)$ ...

**0**

votes

**1**answer

126 views

### p-summable sequence

Let Y be a closed linear subspace of X and suppose that Y does not have copy of l1 .Does each weakly p-summable sequence in X/Y has a subsequence that's the image of a weakly p-summable sequence in X ...

**10**

votes

**3**answers

318 views

### Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters

Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...

**1**

vote

**1**answer

155 views

### Dense subspaces of $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert ...

**11**

votes

**1**answer

279 views

### Complemented subspaces in the dual of James' space $J$

James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...

**1**

vote

**1**answer

143 views

### Hermitian Projections on $C[0,1]$

If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by ...

**1**

vote

**2**answers

134 views

### Antiproximanal subspace of $L_1[0,1]$

Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.
I read somewhere that $Y$ ...

**5**

votes

**1**answer

143 views

### Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows:
Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...

**1**

vote

**0**answers

46 views

### Coersivity of a bilinear form [closed]

I need to proof the coersivity of the following bilinear form. a,b and c are scalars, u is the velocity vector field and p is the pressure. Any help is much appreciated!
$$ B(\textbf{u},\textbf{v}) = ...

**4**

votes

**1**answer

144 views

### A quantity measuring weak non-compactness

Let $A$ be a bounded subset of a Banach space $X$. Set: $wk_{X}(A)=\inf\{\epsilon>0:\overline{A}^{w^{*}}\subset X+\epsilon B_{X^{**}}\}$, where $\overline{A}^{w^{*}}$ denotes the $weak^{*}$ closure ...

**0**

votes

**1**answer

57 views

### The completeness of locally convex space generated by relatively weakly $p$-compact sets

Let $X$ be a Banach space and $1\leq p<\infty$. A bounded subset $K$ of $X$ is relatively weakly $p$-compact if $K$ is contained in $S(B_{l_{p^{*}}})$for some operator $S$ from $l_{p^{*}}$ into ...

**4**

votes

**2**answers

187 views

### Finitely isometrically persistent metric spaces

The goal of this question is to develop further the discussion
initiated in Under which conditions is it possible to find points with same distances under bi-Lipschitz map. The mentioned question was ...

**7**

votes

**1**answer

267 views

### A Hilbert space characterization via retractions--a conjecture

Given a Banach space $X$ and a functional $f:X\rightarrow \mathbb R$, let
$$ X_f := \{x\in X : f(x)\ge 0\} $$
("functional" means "non-zero linear functional"). Also, given a topological space $E$ ...

**9**

votes

**0**answers

215 views

### Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable?

Let $Y$ be a Banach space isomorphic to $\ell_p$, $1<p<\infty$. Is it true that any finite subset of $\ell_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting ...