A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

learn more… | top users | synonyms

5
votes
2answers
139 views

$L^p$ space question

Assume $(X,\mathcal{M},\mu)$ is a measure space and for some $1\leq p<\infty$, $1\leq q<\infty$, $L^p(\mu)\subset L^q(\mu)$. Prove there is a constant $C>0$ so that $\|f\|_q\leq C\|f\|_p$ ...
2
votes
2answers
3k views

Prove that the normed space $L^{\infty}$ equipped with $\lVert\cdot\rVert_{\infty}$ is complete. [duplicate]

Possible Duplicate: Understanding proof of completeness of $L^{\infty}$ Most of the materials I have in Real Analysis consider this statement as a trivial one: "The normed space ...
6
votes
1answer
316 views

Weak-* continuity of the adjoint map on a $W^*$-algebra

Let $\mathcal{M}$ be a $W^*$-algebra, i.e. a $C^*$-algebra with a Banach space predual $\mathcal{M}_*$. I'm trying to show that the adjoint map $x \mapsto x^*$ on $\mathcal{M}$ is weak-* (aka ...
1
vote
1answer
423 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 ...
1
vote
0answers
74 views

How to define operators on $\ell^p_{00}$?

Given $p\in (1,\infty)$. Take a bounded sequence $(f_n)$ in $\ell^p$ and define a linear map $T\colon \ell^p_{00}\to \ell^p$ ($\ell^p_{00}$ is the space of finitelty supported sequences in $\ell^p$) ...
7
votes
2answers
232 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 ...
5
votes
2answers
120 views

The reflexivity of the product $L^p(I)\times L^p(I)$

I am starting to read about the Sobolev spaces $W^{1,p}(I),$ where $I$ is an open interval in $\mathbb{R}.$ In order to establish the reflexivity of $W^{1,p}(I)$ for $p\in ]1,\infty[,$ I need the ...
1
vote
1answer
54 views

Uniformly placed copies of $\ell_1^n$.

Let $\ell_1^n$ denote $\mathbb{C}^n$ endowed with the $\ell_1$-norm. Is it possible that a reflexive space contains isometric copies of all $\ell_1^n$s complemented by projections with norm bounded by ...
4
votes
1answer
134 views

Connected components that are relatively open in $\sigma(T)$

Let $T$ be an bounded linear operator on a Banach space $X$. Suppose the spectrum of $T$, $\sigma(T)$ has infinitely many connected components, then $\sigma(T)$ must contain infinitely many ...
2
votes
1answer
64 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 $C(x,\ep)$ ...
1
vote
1answer
154 views

Immediate predecessor in a chain of subspaces

Let $\mathcal{C}$ be a chain of subspaces of a Banach space $\mathcal{X}$. For each $\mathcal{Y}\in\mathcal{C}$, define its immediate predecessor ...
2
votes
1answer
70 views

Perturbing an abstract discrete dynamical system

Let $X$ be a Banach space. Denote for $x_0\in X$ and $r>0$ the closed ball centered at $x_0$ by $B(x_0,r)=\lbrace x\in X:\|x-x_0\|\le r\rbrace$. Suppose $f:X\to X$ a bounded map with a fixed point ...
8
votes
1answer
423 views

Linear contraction on a Banach space

Let $X$ be a Banach space with a norm $\|\cdot\|_1$ and $A$ be a linear operator on $X$ such that $\|A\|_1\leq 1$; $\|A^m\|_1<1$ for some $m\in \mathbb N$. Is that true that there is an ...
2
votes
1answer
329 views

Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
1
vote
1answer
102 views

Operators from $\ell_p$ to $\ell_q$

Let $1\leqslant p<q<\infty$. Denote $L(\ell_p)$ the space of bounded operators on $\ell_p$. Let $B_{L(\ell_q)}$ [was $B_{L(\ell_p)}$ ] be the closed unit ball of $L(\ell_q)$ [was $L(\ell_p)$] ...
2
votes
1answer
218 views

WOT closure and SOT closure of convex sets

I am reading some papers on operators acting on Banach spaces and one of them uses the following fact: If a vector space has two locally convex topologies with identical collections of continuos ...
2
votes
1answer
102 views

Almost invariant subspaces for WOT closure of an algebra of operators

Let $X$ be a Banach space and $\mathcal{C}\subset\mathcal{L}(X)$ be a collection of bounded linear operators. A subspace $Y$ is said to be almost invariant under $\mathcal{C}$ if for each ...
152
votes
0answers
5k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
4
votes
1answer
1k views

Question about Fredholm operator

$X,Y$ are Banach spaces and $A\in B(X,Y)$ is a Fredholm operator (that is, the dimensions of ker($A$) and coker($A$) are both finite), then are closed linear subspaces ker($A$) and Im($A$) ...
8
votes
1answer
292 views

Subspaces of $l_p$ and Banach-Mazur distance

It is well-known that every subspace of $l_2$ is isometric to $l_2$. When $p\neq 2$, $l_p$ has subspaces that are not even isomorphic, let alone isometric, to $l_p$. Suppose $X$ is a subspace of $l_p$ ...
3
votes
4answers
1k views

Gâteaux derivative

Let $X$ be a Banach space and $\Omega \subset X$ be open. The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle ...
2
votes
0answers
261 views

A problem with $\ell_p$-norm

Let $1<p<\infty$ be fixed. Suppose $L=\{(x_1,\dots,x_n):x_i\ge 0, \sum_i x_i=1, \sum_i a_i x_i=b\}$ for some real numbers $a_i$ and $b$. I am wondering whether the following would be true. ...
0
votes
1answer
116 views

Are there simple examples of Banach spaces with no non-trivial Clifford Isometries?

By a Banach space $X$ I mean, a complete normed vector space and by a Clifford isometry I mean a surjective isometry $\gamma$ of $X$ such that the distance $d(\gamma x, x)$ is constant on $X$. ...
1
vote
1answer
178 views

Compact, bounded sets and measures of non-compactness

Let $\gamma$ denote the Hausdorff/Kuratowski measure of noncompactness defined on a Banach space $(X,\|\cdot\|)$. I was wondering whether $\gamma(A)=\gamma(A+K)$ holds for $A\subset X$ is bounded and ...
2
votes
1answer
238 views

Linear functionals on Banach spaces

The following is a homework problem (I have solved the majority of it, but need help with the last part) Suppose $f$ : $V \rightarrow \mathbb{F}$ is a non-zero continuous linear functional on a ...
3
votes
2answers
87 views

Translating subsets in normed spaces

Let $X$ be a Banach space and endow the space $BC(X)$, the space of all bounded closed subsets of $X$, with the Hausdorff distance $d_H$. Fix $C_0\in BC(X)$. Is it true that ...
4
votes
3answers
91 views

$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?

Why does the following hold for continuous functions on $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
2
votes
3answers
429 views

$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$

I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous. Any ideas?
6
votes
3answers
151 views

There exists a unique function $u\in C^0[-a,a]$ which satisfies this property

The problem: Let $a>0$ and let $g\in C^0([-a,a])$. Prove that there exists a unique function $u\in C^0([-a,a])$ such that $$u(x)=\frac x2u\left(\frac x2\right)+g(x),$$ for all $x\in[-a,a]$. ...
0
votes
1answer
79 views

Construct locally lipschitz map from a bounded one

Let $X$ be a Banach space and $BC(X)$ the space of all bounded closed subsets in $X$. It can be shown that $(BC(X),d_H)$ is a complete metric space (see this page for a definition of $d_H$). If ...
2
votes
1answer
143 views

Power bounded operators

Let $X$ be a separable reflexive Banach space and let $T$ be a power-bounded operator on $X$ ($\sup_n \|T^n\|<\infty$.) Let $S$ be a WOT-limit point of $(T^n)$. Suppose for some $n$ we have ...
3
votes
2answers
137 views

completeness of cones in an ordered normed space

Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
3
votes
2answers
267 views

Dense subspaces in complete TVS

If $X$ is a complete topological vector space, Y is a dense subspace (so $\overline{Y}=X$), Z is a closed subspace, it is possible that $Y\cap Z=\{0\}$? This is definitely possible for subsets in ...
2
votes
1answer
199 views

How to describe the space $L_{\infty}(\mu,X)$?

Given a Banach space $X$ and a measure space $(\mathfrak{A}, \mu)$ One can form the Banach space $L_\infty(\mu, X)$ of all measurable, essentially bounded functions from $\mathfrak{A}$ to $X$. Is it ...
1
vote
1answer
93 views

Characterization of strong minimums with slices.

I am doing a proof of a Lemma that isn't in a book. Let $X$ a Banach space and $\emptyset\not=S\subset X$ closed of $X$. Let $f$ be a lower semicontinuous function bounded below in $S$. I have that ...
19
votes
1answer
1k views

Cardinality of a Hamel basis

What is the cardinality of a Hamel basis of $\ell_1(\mathbb{R})$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant ...
1
vote
1answer
102 views

Prove that the space is not complete

Let $X$ be a separable space with infinite dimension, let $(\cdot,\cdot)$ and $\|\cdot \|$ be the scalar product and the norm of $X$, and $\{e_n\}_n$ be an orthonormal basis of $X$. We define ...
1
vote
1answer
252 views

Strongly exposed points/Exposed points

I was studying and I got the next doubt: We suppose that $(X,\|\cdot\|)$ is a Banach space and $C$ it is a convex closed subset of X. We say that $x\in C$ it is an exposed point of $C$ if $\exists ...
6
votes
1answer
469 views

Finite dimensional subspaces

Let $X$ be a complex Banach space of infinite dimension. Does there exist a finite dimensional subspace of $X$ of arbitrary (finite) dimension which is complemented by a projection of norm 1?
6
votes
1answer
113 views

Bounded extension

What are the easiest examples of a pairs of Banach spaces $X,Y$ such that $X\subseteq Y$ ($X$ is a closed linear subspace of $Y$) there is a bounded linear map $T\colon X\to Y$; there is no bounded ...
4
votes
1answer
259 views

How to prove that there is a subspace $W \subset C(X)$ so that $C(X)$ is isomorphic?

Let $X$ be a compact metric space and let $F$ be a closed subset of $X$. Assume that there exists a bounded extension operator $T:C(F) \rightarrow C(X)$, i.e., $T \in B(C(F),C(X))$ and for all $g\in ...
3
votes
1answer
608 views

For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.

I want to prove that for Banach space V there is a compact topological space $X$ so that $V$ is isometrically isomorphic to a closed subspace of $C(X)$-continuous function on a (compact) topological ...
4
votes
1answer
374 views

what exactly is weak* topology?

I know that weak* topology is the weakest topology so that $Jx$ is continuous for $\forall x\in X$, where $J$ is the isometry from $X$ to $X''$. But what exactly is this topology? What is the open set ...
8
votes
1answer
1k views

What is the predual of $L^1$

Is there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a space. How do you start to find such preduals ...
14
votes
1answer
1k views

Direct aproach to the Closed Graph Theorem

In the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, and ...
8
votes
1answer
459 views

Is this operator compact?

Suppose ($x_n$) is a normalized, linearly independent, sequence in a reflexive Banach space $X$, and $T$ is an injective, strictly singular, bounded operator on $X$ such that $Tx_n\longrightarrow ...
3
votes
1answer
85 views

comparison between spaces

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of ...
2
votes
1answer
267 views

Example of strictly convex space with not strictly smooth dual

I'm trying to find an example of a space $V$ which is strictly convex, but has a dual space $V^*$ which is not strictly smooth. Any help please?
1
vote
2answers
569 views

Convex hull of the extreme points of the unit ball in $C(K)$ where $K$ is the Cantor set.

Let $K$ denote the usual $1/3$ Cantor set and let $B=B_{C(K)}$ (here $B_{C(K)}$ = {$v \in C(K) : \|v\| \leq 1 $} denotes the closed unit ball of $C(K))$. Then how to prove that $B$ coincides with the ...
8
votes
2answers
836 views

Complement of $c_{0}$ in $\ell^{\infty}$

How can I show that $c_{0}$ cannot be complemented in $\ell^{\infty}$? Complement in the following sense $$c_{0}+V = \ell^{\infty}$$ And the projections are continuous.