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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A convergence problem in Banach spaces related to ergodic theory

Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition. $\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$ $\frac{1}{n}\lVert ...
8
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3answers
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A question about Banach reflexive space

How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?
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77 views

The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).

Let $A\in B(\mathbb{C}^n) \cong \mathbb{M}_n(\mathbb{C} )$ Prove: The matix of $A^*$ (Banach space adjoint of $A$) with respect to the dual basis coincides with $A^t$ (the transpose of $A$).
13
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1answer
346 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
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2answers
142 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 ...
3
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1answer
136 views

A problem in Banach space

X is Banach space, $f_{i} \in X^{*},\forall 1\leq i<\infty$ and $ \sum_{i=1}^{\infty}| f_{i}(x)|<\infty,\forall x\in X$ how to prove the following lemma: $\forall F \in X^{**},\quad ...
2
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1answer
155 views

Subadditive, submultiplicative and scalar-multiplication-invariant functions

Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...
4
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1answer
97 views

For a reflexive Banach space, we have $\left\Vert x-y\right\Vert =\left\Vert x+F\right\Vert _{E/F}$

The problem is: Let E be a Banach space and $F\subset E$ be a closed linear subspace. Prove that for every $x \in E$ there exists $y \in F$ such that $\left\Vert x-y\right\Vert =\inf\left\{ ...
1
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1answer
459 views

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different ...
5
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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
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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
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1answer
308 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
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1answer
414 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 ...
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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
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2answers
227 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
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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
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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
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1answer
133 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
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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 $C(x,\ep)$ ...
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1answer
153 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
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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 ...
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1answer
412 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
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1answer
321 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 ...
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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
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1answer
211 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
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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 ...
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0answers
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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 ...
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1answer
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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($A$) ...
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290 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 $l_p$ ...
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4answers
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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 ...
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0answers
260 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
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1answer
115 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 $X$. ...
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1answer
176 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
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1answer
235 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 ...
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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
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3answers
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$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ 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
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3answers
426 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?
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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
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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 ...
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1answer
142 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 ...
2
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1answer
124 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 ...
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2answers
261 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 ...
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1answer
198 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 ...
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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
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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 ...
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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 ...
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1answer
245 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 ...
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1answer
466 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
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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
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1answer
258 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 ...