2
votes
0answers
32 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
0
votes
0answers
20 views

Summability of Fourier series from Banach space point of view

I am under the impression the following is true (any pointer to a reference would be appreciated ): Theorem (Katznelson?) For any $f \in C[0,1]$ with Fourier coefficients $\{ \hat{f}(n)\}$, there ...
3
votes
0answers
40 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
5
votes
0answers
117 views

Reference request for the fact

Does anyone know a reference to the paper or a textbook where this fact is proved $$ \mathcal{B}(\bigoplus_1 X_\alpha, Y)\cong_1 \bigoplus_\infty \mathcal{B}(X_\alpha, Y) $$ Most author are bored to ...
1
vote
1answer
38 views

References for Banach Space Theory

Currently I am investigating the proof of Banach-Stone Theorem.. I would like to strengthen my background in Banach Spaces so that I can fully understand the proof. I am using this to study the proof. ...
1
vote
3answers
205 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
3
votes
1answer
128 views

Frechet derivative of compact operator is compact

... this seems to be a well known fact as mentioned in this and in this manuscript. However, I was not able to find a proof or to prove it by myself. So my question is: How to prove this? Any hint ...
0
votes
1answer
30 views

Do monotone operators have positive Frechet derivatives?

If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$. Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces ...
1
vote
0answers
71 views

Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
1
vote
1answer
185 views

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like All classical (separable) Banach spaces used in analysis have a Schauder basis . I was wondering where to find a proof confirming ...
3
votes
1answer
69 views

Extend a linear functional on “nice” functions to an integral

I have a positive linear functional $h$ defined on a set of Lesbesgue-measurable functions of "moderate growth" on $\mathbb{R}^2$–call this set $MG(\mathbb{R}^2)$. (A function $f$ is positive if ...
9
votes
0answers
221 views

Infinite dimensional constant rank theorem

Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
1
vote
1answer
113 views

Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite ...
3
votes
1answer
180 views

Morrey space and Campanato space.

I'd like to know a lot about Morrey space and Campanato spaces. For example, I'd like to know how can I see the details presents here. I'd like some reference about this. I thank you very much.
1
vote
0answers
42 views

geometric charaterization of complex interpolation spaces $(H,Y)_\theta$ where $H$ is a Hilbert space?

Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that $$ X=(H,Y)_\theta $$ (complex interpolation of Calderon). ...
0
votes
1answer
94 views

Relationship between different$L^p (\Omega, \mathcal{F}, \mu)$ spaces with $\Omega$ uncountable and $\mu$ being a Radon measure

I earlier asked this question but I have not had a general classification in the posted answers there. So here is a new question. I am looking now for some special cases as suggested in one of the ...
0
votes
3answers
160 views

References on relationships between different $L^p$ spaces

I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
8
votes
1answer
385 views

$C_0(X)$ is not the dual of a complete normed space

Let $X$ be any locally compact Hausdorff space and assume that it is not compact. I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of ...
2
votes
1answer
264 views

Are compacta in a complete infinite dimensional normed space nowhere dense?

Let $X$ be an infinite dimensional Banach space. I want to show that any compact subset $\varnothing\neq A\subset X$ is nowhere dense. I've been able to prove the statement for ...
5
votes
2answers
362 views

How to prove that the $L^p$ spaces are infinite dimensional

It is well-known that (given a measure space $(S,\mathcal A,\mu)$ and $1\le p\le\infty$) the Banach space $L^p(S,\mathcal A,\mu)$ has infinite dimension. Is there an easy way to proof this statement ...
1
vote
1answer
91 views

Where to find information on the Hilbert cube in $\ell^2$

The Hilbert cube $H$ in $\ell^2=\ell^2(\mathbb{R})$ is the subset given by $$H=\lbrace(x_n)=(x_1,x_2,\ldots)\in\ell^2:|x_n|\le2^{-n} \text{ for }n=1,2,\ldots\rbrace.$$ I've heard that ...
1
vote
1answer
131 views

norm of operator in Hilbert space and complex conjugate Banach space

Let $E$ and $F$ be complex Banach spaces. We denote by $\overline{E}$ the compex conjugate of $E$, that is, the vector space $E$ with the same norm but with the conjugate multiplication by a complex ...
1
vote
2answers
521 views

Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces

Looking for the proof of Eberlein-Smulian Theorem. Searching for the proof is what I break with this morning. Some of my friends recommend Haim Brezis (Functional Analysis, Sobolev Spaces and ...
13
votes
1answer
323 views

How does $\sigma(T)$ change with respect to $T$?

Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane. I wonder whether there is some result concerning how ...
0
votes
2answers
127 views

On Reflexive Banach Spaces

My Functional Analysis lecturer gave me a topic for my assignment, the title is "On Reflexive Banach Spaces". I am a looking for several good references to start my work, that is why I brought this ...
1
vote
1answer
274 views

Nearest point projection in uniformly convex Banach spaces

Let $X$ be a uniformly convex Banach space, $x\in X$ and $C\subset X$ closed and convex, then there is a unique $y\in C$ with $$\lVert x-y\rVert=\inf_{z\in C}\lVert x-z \rVert.$$ Is there a good book ...
7
votes
2answers
181 views

Spectra of restrictions of bounded operators

Suppose $T$ is a bounded operator on a Banach Space $X$ and $Y$ is a non-trivial closed invariant subspace for $T$. It is fairly easy to show that for the point spectrum one has ...
2
votes
1answer
55 views

An abstract $\alpha$-contracting dynamical system

$\newcommand{\f}{\phi}$$\newcommand{\ep}{\varepsilon}$$\newcommand{\R}{\mathbb R}$ Suppose $(\f_t)_{t\ge0}$ is an abstract dynamical system in a Banach space $(X,\|\mathord\cdot\|)$. Let $C(x,\ep)$ ...
21
votes
0answers
1k views
+50

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
32
votes
2answers
1k views

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$? In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
5
votes
2answers
294 views

Banach-Saks property and reflexivity

On the German Wikipedia page on the Banach-Saks property, they claim that every Banach space with the Banach-Saks property is reflexive but that the converse is not true. There should be a proof due ...
0
votes
1answer
67 views

reference for conditional expectation

Suppose $1\leq p<\infty$. Let $E$ be a Banach space. Consider a filtration $F_n$ on some probability space $\Omega$. Let $X\in L^p(\Omega,E)$ where $L^p(\Omega,E)$ denote the Bochner space. In ...
4
votes
1answer
120 views

Bidual of a WSC space

Let $E$ be a Banach space which is weakly sequentially complete (i.e. each weak Cauchy sequence converges weakly). Must $E^{**}$ be weakly sequentially complete either? Of course, this question is ...
5
votes
1answer
745 views

Reflexive space which is not uniformly convex

I found this beautiful theorem (Milman-Pettis): Every uniformly convex Banach space is reflexive I think it's a remarkable statement, since uniformly convexity is a geometric property of the norm ...
2
votes
1answer
109 views

Is a Fréchet differentiable map between complex Banach spaces locally given by a “power series”?

Let $X,Y$ be Banach spaces over $\mathbb{C}$ and let $U \subset X$ be open. If $f:U \to Y$ is Fréchet differentiable at every point of $U$, can we locally expand $f$ as a "power series"? To be more ...
10
votes
1answer
356 views

Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...
5
votes
1answer
152 views

Comparison between Rademacher average and random average

Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of ...
14
votes
4answers
358 views

Banach spaces over fields other than $\mathbb{C}$?

Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...
3
votes
1answer
196 views

weak$^*$-separability of $l_\infty^*$.

Where can I find the proof that $l_\infty^*$ is weeak$^*$-separable? I want to re-examine the proof of that fact.
6
votes
3answers
268 views

$L^1$ space with values in a Banach Space

I have been reading a bit about the Bochner integral and now I'm wondering the following: For the theory to be "nice", one would expect that $$L^1([0, \tau], L^1([0, \tau])) \cong L^1([0,\tau] ...