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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An explicit example of an invariant halfspace of the unilateral shift?

In a recent talk, A. Popov stated the following fact The unilateral shift on $\ell^2$ has invariant halfspaces. Halfspaces are closed subspaces whose dimension and codimension are both infinite. ...
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166 views

The separation theorem for the w* topology

When I read the prove of the Goldstine Theorem(See An Introduction to Banach Space Theory Robert E. Megginson 2.6.26), I find it use the separation theorem for the w* topology without any details. ...
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60 views

Convergent operatorial series

An exercise I was doing asks (among other things) for the values of $z\in\mathbb{C}$ for which the following (operatorial) series converges absolutely: $$\sum_{n=0}^{\infty}z^nA^n$$ where $A$ is an ...
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158 views

Banach space, Normed vector space

Help me please with this question. Let's $Y$ be Banach space, $Z$ - Normed vector space and $(T_{n})_{\mathbb{N}}$ - the sequence in $B(Y,Z)$ so that all sequence $(y_{n})_{\mathbb{N}}$ in Y holds: ...
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179 views

Dual without quotients isomorphic to $c_0$

The following question is bothering me. Suppose we have a dual Banach space $X^*$ and assume $X^*$ has a quotient isomorphic to $c_0$. Must $X^*$ contain a complemented copy of $\ell_1$?
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243 views

Criterion for a limit of invertible operators on a Banach space to be invertible

Let $A_n$ linear operators in a Banach space $B$ that have inverses. $||A_n-A|| \to 0$ for some operator $A$. I need to prove that $A$ has an inverse operator iff the sequence $\{||A_n^{-1}||\}$ is ...
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191 views

Embedding dual of a Banach space into a predual

Let $X$ and $Y$ be Banach spaces and suppose moreover that there is an isometric embedding of $X^{**}$ into $Y$. Assume moreover that $Y$ has the unique predual $Y_*$ up to isometry (like von Neumann ...
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711 views

Must a weakly or weak-* convergent net be eventually bounded?

Let $\mathfrak{X}$ be a Banach space. As a standard corollary of the Principle of Uniform Boundedness, any weak-* convergent sequence in $\mathfrak{X}^*$ must be (norm) bounded. A weak-* convergent ...
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1answer
173 views

An upper bound for $\|(\lambda-A)^{-1}\|$?

Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$. If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...
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489 views

Is the complement of a finite dimensional subspace always closed?

Let $F$ be a finite dimensional subspace of an infinite dimensional Banach space $X$, we know that $F$ is always topologically complemented in $X$, that is, there is always a closed subspace $W$ such ...
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489 views

Finding the topological complement of a finite dimensional subspace

I know that for any finite dimensional subspace $F$ of a banach space $X$, there is always a closed subspace $W$ such that $X=W\oplus F$, that is, any finite dimensional subspace of a banach space is ...
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149 views

How to determine the $\delta$ in the open mapping theorem?

Let $X$ and $Y$ be Banach spaces and $T\in\mathcal{L}(X,Y)$ be a bounded linear operator from $X$ to $Y$. If $T$ is surjective, then the open mapping theorem says that there is a positive $\delta$ ...
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245 views

How to show that these spaces are Banach spaces

I want to show, that the following spaces are Banach spaces: $X_1:=\{M=(M_t)_{0\le t \le T} ;\mbox{ M is an adapted RCLL process }\}$ with the norm $\|M\|_{X_1}:=\|\sup_{0\le t\le T}|M_t|\|_{L^2(P)}$ ...
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629 views

Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = ...
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186 views

countable intersection of closed convex bounded subsets reflexive banach space is non empty.

If $X$ is a reflexive Banach space and $(C_n), n \in \mathbb{N}$ is a sequence of closed convex bounded sets with $C_{n+1}$ contained in $C_n$ for all $n \in \mathbb{N}$. How does one show that the ...
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71 views

When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
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142 views

Is $C^1(A)$ a Banach space?

Let $A \subset \mathbb R$ and consider the space $C^1(A)$. I am asked to prove that $( C^1(A), \Vert \cdot \Vert_{C^1(A)})$ is a Banach space, where $$ \Vert f(x) \Vert_{C^1(A)} = \sup_{x \in A} ...
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84 views

How to Prove ($\mathbb{C}\langle x, y \rangle$, $\|\cdot\|$) is a Banach Space

Let $\mathbb{C}\langle x,y\rangle$ be the group ring of the complex numbers over the free group in $x,y$. Let $len : \langle x,y \rangle \rightarrow \mathbb{N}$ denote the standard word norm and let ...
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267 views

Operators from $\ell^\infty$ into $c_0$

I have the following question related to $\ell^\infty(\mathbb{N}).$ How can I construct a bounded, linear operator from $\ell^\infty(\mathbb{N})$ into $c_0(\mathbb{N})$ which is non-compact? It is ...
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131 views

What is the dual of the space L-infinity ($L^\infty$)? [duplicate]

Possible Duplicate: The Duals of $l^\infty$ and $L^{\infty}$ In learning real analysis, I do understand that the dual of $L^\infty$ cannot be $L^1$ because the latter is separable, whereas ...
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1answer
438 views

What is the precise definition of predual

How does one define "predual" and the surrounding notions? More specifically: Why must there be only one predual of $X$ when $X$ is a Banach space? What is the correct notion of similarity here ...
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120 views

Space of $\mathbb{R}$ valued sequences converging to $0$. Some basic results.

Let $C_0(\mathbb{R})$ be the space of $\mathbb{R}$ valued sequences converging to $0$. Let $l_n$ be a positive sequence in $\mathbb{R}$ such that $\sum\limits_{n=1}^\infty l_n=1$. We define $$ ...
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Example of non-reflexive Banach space

How does one prove that $C^0([0;1],\mathbb{R})$ equipped with the sup norm is not reflexive? I don't understand how to show that the $J$ mapping is not surjective.
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continuous linear functional on a reflexive Banach space attains its norm

How does one prove that if a $X$ is a Banach space and $x^*$, a continuous linear functional from $X$ to the underlying field, then $x^*$ attains its norm for some $x$ in $X$ and $\Vert x\Vert = 1$? ...
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855 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 ...
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150 views

A subset of $\bar{S}\backslash S$ contains an open ball in $\bar{S}$? (operator theory)

E and S are subsets of a metric space. $E$ is a subset of $\bar{S}\backslash S$. Then $\overline{E}\subset(\overline{S}\backslash S^{o})$, but I wonder whether there is some condition that guarantees ...
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164 views

When a projection is an adjoint of another operator

Suppose we have a Banach space $X$ and an idempotent operator $Q\colon X^*\to X^*$ with range isomorphic to $\ell_1$. Must $Q$ be an adjoint to some idempotent operator on $X$? In other words, is $Q$ ...
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84 views

Least fixpoint in a Banach space of bounded measurable functions

Let $(E,\mathscr E)$ be a measurable space and denote by $\mathrm b\mathscr E$ the space of all Borel measurable bounded functions $f:E\to\mathbb R$. On this space the partial order is given by $$ ...
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53 views

What does this statement mean? $C^k (I, E)$

$C^k (I, E)$ := space of $k$ times differentiable functions from an interval $I$ into a Banach space $E$. I don't know the exact meaning of "into a Banach space". Please help me.
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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 ...
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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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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 $A$).
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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 ...
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144 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 ...
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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 ...
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1answer
156 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}, ...
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98 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\{ ...
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466 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 ...
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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 $\|f\|_q\leq C\|f\|_p$ ...
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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 ...
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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 ...
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428 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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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 $\ell^p$) ...
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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 ...
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121 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 ...
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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 ...
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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 ...
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65 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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157 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 ...
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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 ...