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

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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
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598 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
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1answer
369 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 ...
7
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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
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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
458 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
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1answer
265 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?
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2answers
563 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
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2answers
805 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.
5
votes
1answer
537 views

Exhibiting open covers with no finite subcovers.

How do I exhibit an open cover of the closed unit ball of the following: (a) $X = \ell^2$ (b) $X=C[0,1]$ (c) $X= L^2[0,1]$ that has no finite subcover?
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vote
1answer
162 views

How to prove that the inverse exists on the whole space X?

I was reading Kreyszig's book on functional analysis when I came across this theorem: "Let $T\in{}B(X,X)$, where $X$ is a Banach space. If $||T||<1$, then $(I-T)^{-1}$ exists as a bounded linear ...
2
votes
1answer
447 views

Convex hull of unions

I would like to find a representation for convex hulls co$(\cdot)$ (see wikipedia for the definition of the convex hull) in normed spaces. Let $A,B\subset X$ be bounded and convex subsets of a normed ...
3
votes
2answers
2k views

Prove that $X'$ is a Banach space

I'm taking a new course on functional analysis and meet with the following problem. If $X$ is a normed space (not necessarily complete), then prove that $X'$ is a Banach space. Definition: When the ...
7
votes
1answer
281 views

Expectation Problem (Khintchine's Inequality)

As a reference, this is Problem 6.2 in Albiac and Kalton's Topics in Banach Space Theory The question involves a direct proof of Khintchine's Inequality. In part (1), we are to prove that ...
2
votes
1answer
212 views

Adjoint identity

I want to show that $\operatorname{Range}(A^*)^\perp \subset \operatorname{Null}(A)$ where $A:E \supset D(A) \to F$ is an unbounded closed linear operator densely defined in $E$, and $E$ and $F$ are ...
5
votes
2answers
456 views

Isomorphism of Banach spaces implies isomorphism of duals?

I can't make up my mind whether this question is trivial, or simply wrong, so i decided to ask, just in case someone sees a fallacy in my reasoning: Question: Suppose $V,W$ are two banach spaces, and ...
4
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0answers
67 views

complemented subspaces of $L_{p}$ spaces (Question posed incorrectly earlier)

This question was asked incorrectly originally in such a way that it probably made no sense. Fixed version: I know that $L_{p}[0,1]$ has $\ell_{2}$ as a complemented subspace, and I'm wondering if ...
5
votes
1answer
212 views

Boolean algebras of projections

Suppose $X$ is a Banach space with an unconditional basis $(e_n)$. Then, one may easily define a Boolean algebra of projections in $\mathcal{L}(E)$ which is isomorphic to the power-set of $\mathbb{N}$ ...
4
votes
1answer
815 views

Banach space in functional analysis

Prove that a closed subspace of a Banach space is also a Banach space. Show that the linear space of all polynomials in one variable is not a Banach space in any norm.
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1answer
287 views

Show sequence equicontinuous

I don't know how to prove this question: Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
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votes
2answers
403 views

Totally bounded space

Suppose $M=\left \{ f\in L^1([0,1])\, |\, 0<f(x)<\frac 1{\sqrt x} \text{almost everywhere on} \, (0,1) \right\}$. Is it true or not, that $M$ is totally bounded?
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0answers
2k views

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 ...
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1answer
576 views

$\ell_\infty$ a Grothendieck space

The problem I am considering stated formally is this: Show that if a sequence in $\ell_\infty^*$ is weak*-convergent, then it is also weakly convergent. We may reduce this to the case where the ...
4
votes
1answer
193 views

There exists an isometric embedding

Let $W$ be a closed linear subspace of a normed vector space $V$. Let $i_V: V \to V^{**}$. and $i_W: W \to W^{**}$ be the canonical embeddings of V and W into their second duals. Prove that there ...
4
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1answer
571 views

space of bounded measurable functions

Let $(\Omega, \Sigma)$ be a measurable space. Is the space of bounded measurable functions $B_b(\Sigma)$ equipped with the supremum norm a Banach space, i.e. complete?
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1answer
137 views

Hilbert space $H$ is strictly smooth

I am trying to show that every Hilbert space $H$ is strictly smooth with modulus of smoothness $\phi_H(t)=\sqrt{1+t^2} -1 $. To show this I think I should show $H$ is uniformly smooth first. ...
7
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1answer
762 views

The group of invertible linear operators on a Banach space

Let $X$ be a Banach space. Let $G$ be the group of invertible linear operators from $X$ to itself. Now my questions are: If $G$ is equipped with the operator norm topology, how do you show that it ...
3
votes
1answer
204 views

$\ell_1$ dense in $c_0$?

This may be a silly question, but here goes. To ensure clarity, $\ell_1$ is the space of absolutely summable sequences, and $c_0$ the space of bounded sequences with limit 0. So we know that ...
3
votes
0answers
146 views

What is the Dunford Integral and why is it useful?

Wikipedia defines the Pettis Integral for Banach space valued functions on a measure space by duality. Apparently there is a Dunford integral which specializes to the Pettis integral. What is its ...
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104 views

Operators with complemented range

Holub proved that Fredholm operators are stable under compact perturbations. I am interested in a slight refinement of this theorem. Suppose we have two operators $T_1$ and $T_2$ acting on a primary ...
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1answer
89 views

A question regarding convergence of distances to closed balls in Banach spaces

Let $X$ be Banach and let $B(x,\varepsilon)$ be the closed ball of radius $\varepsilon>0$ around $x\in X$ and consider the sequence $$f_{n;x}(y)= \begin{cases} 1-n\cdot d(yB(x,\varepsilon)), ...
4
votes
1answer
186 views

Measuring closed balls

Let $(X,\parallel \cdot \parallel)$ be Banach and $$\mathcal{BC}(X)=\{A\subset X\colon A \text{ is closed, bounded and non-empty}\}.$$ The natural metric on this space is the Hausdorff distance $d_H$ ...
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vote
1answer
123 views

Convergence of a sequence of functions on closed balls

Let $X$ be a Banach space and $d$ be the induced metric. Let $S(x;r)$ denote the closed ball with radius $r$ at $x\in X$, that is,$$S(x;r)=\lbrace y\in X\colon d(x,y)\le r\rbrace.$$ Let $x,y\in X$ ...
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2answers
778 views

Linear functional on a Banach space is discontinuous then its nullspace is dense.

I need to prove that: If a nonzero linear functional $f$ on a Banach Space $X$ is discontinuous then the nullspace $N_f$ is dense in $X$. To prove that $N_f$ is dense, it suffices to show that ...
6
votes
2answers
360 views

Definitions of measurability for operator-valued functions

If $\mathfrak{X}$ is a Banach space, a function $T: \mathbb{R} \to \mathcal{L}(\mathfrak{X})$ is defined to be uniformly measurable if it is an a.e. norm limit of a sequence of countably valued ...
1
vote
1answer
249 views

Quotient space is isometrically isomorphic to $C(F)$

Let $X$ be a compact metric space and let $F$ be a closed subset of $X$. If $W = \{f \in C(X) : f(x) = 0 \text{ for all } x \in F\}$, show that $C(X)/W$ is isometrically isomorphic to $C(F)$. ($C(X)$ ...
0
votes
1answer
645 views

Bounded linear transformation

For a compact metric space $X$, $C(X)$ denotes the space of continuous real-valued functions on $X$ equipped with the supremum norm. Let $X$ and $Y$ be compact metric space and let $g:X \to Y$ be a ...
6
votes
1answer
201 views

Is this set corresponding to a bounded linear operator necessarily open?

Let $\Lambda : X \to X$ be a bounded linear operator on a Banach space $X$. My question is whether the set $$ \{\lambda \in \mathbb C: \lambda I - \Lambda \quad\text{is surjective} \} $$ is ...
5
votes
1answer
206 views

$A\oplus B\cong A\oplus C$ implies $B\cong C$? (No, it does not)

I am asked to prove that for $p\in (1, \infty)$, $$L_{p}[0,1]\cong L_{p}[0,1]\oplus \ell_{2}$$ on a homework assignment, and I think I can show using results from class that $\ell_2\oplus \ell_2\cong ...
4
votes
1answer
146 views

Image of a Banach space

Let $X$, $Y$, and $Z$ be Banach spaces with $Z \subset X$. Suppose $T$ is a bounded linear operator with domain $X$ and range $Y$. Must $T(Z)$ be a Banach space?
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2answers
477 views

$\ell_1$ and unconditional convergence

Thanks to the Riemann theorem we know that absolute convergence and unconditional convergence are the same for $\mathbb{R}$. In all the Frechet spaces absolute convergence implies unconditional ...
5
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1answer
419 views

If $L$ is normed and for every hyperplane $M \cap \operatorname{ball} L^*$ weak*-closed implies $M$ weak*-closed then $L$ is a Banach space

If $L$ is a normed space with the property that if $M$ is a hyperplane in $L^*$ and $M \cap \operatorname{ball} L^*$ is weak-star closed $\implies$ $M$ itself is weak star closed, then how do I show ...
4
votes
3answers
637 views

Cesàro operator is bounded for $1<p<\infty$

The Cesàro operator $T\colon \ell_{p}\to\ell_{p}$ is defined by $(Tx)_{k}=\frac{1}{k}\sum_{j=1}^{k}x_{j},\: k\in\mathbb{N}$, where $x=(x_{k})_{k=1}^{\infty}$ Show that $T$ is bounded if ...
2
votes
1answer
328 views

Hahn-Banach. Extend the functional by continuity

Let $E$ be a dense linear subspace of a normed vector space $X$, and let $Y$ be a Banach space. Suppose $T_{0}\in\mathcal{L}(E,Y)$ is a bounded linear operator from $E$ to $Y$. Show that $T_{0}$ can ...
1
vote
1answer
165 views

Sum of bounded and unbounded operators

Is there a Banach space $X$, $S$ an unbounded operator defined on a dense subspace $D$ of $X$ and a bounded operator $T$ on $X$ such that $$S+T|_D$$ is bounded? What if $T$ is assumed to be ...
9
votes
1answer
333 views

An application of J.-L. Lion's Lemma

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot\|_X$, $\|\cdot\|_Y$,$\|\cdot\|_Z$. Assume that $X\subset Y$ with compact "injection" and that $Y\subset Z$ with continuous injection. Then ...
12
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1answer
276 views

Banach spaces isomorphic to square

This is another exercise from Allan's book "Introduction to Banach Spaces and Algebras". Exercise 2.9: A Banach space $E$ is said to be homeomorphic to its square if $E\oplus E$ is linearly ...
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1answer
215 views

$\ell_p$ sums of Banach spaces

Let $p\in (1,\infty)$ and let $(E_\alpha)_{\alpha<\omega_1}$ be a family of Banach spaces. Set $E=\left(\bigoplus_{\alpha<\omega_1}E_\alpha\right)_{\ell_p(\omega_1)}$. Must $E$ be isomorphic to ...
6
votes
1answer
349 views

Is kernel a complemented subspace

Let $\mathcal{A}:X\to Y$ be continuous linear operator, $X$ and $Y$ are Banach spaces. Let $\text{Im} \mathcal{A}=Y$. Is $\ker\mathcal{A}$ a complemented subspace of $X$?