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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Does weak*-separability of $X^*$ imply that $X$ is separable? [duplicate]

Possible Duplicate: Does separability follow from weak-* sequential separability of dual space? $\omega^*$-separability of $l_\infty^*$. Recently I read a Theorem stating, Let $X$ be a ...
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71 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
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73 views

Complementation of $\ell_1$ in dual spaces [duplicate]

Possible Duplicate: Weakly compact operators on $\ell_1$ There is another question I would like to ask, if you don't mind, which is not very far from my previous one. Suppose we have a ...
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459 views

Weakly compact operators on $\ell_1$

Is the following assertion true/known? Let $V$ be a Banach space and let $T\colon \ell_1\to V$ be a bounded linear operator. Is it true that $T$ is not weakly compact if and only if there is a ...
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397 views

Is an open linear map closed (to some extent)?

Suppose we have a surjective bounded linear operator acting between Banach spaces. By the Open Mapping Theorem it maps open sets in the domain to open sets in the codomain. Must the image of a closed ...
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622 views

An application of Riesz' Lemma

How does one prove using Riesz' Lemma that an infinite dimensional subspace $Y$ of a Banach space $X$ contains a sequence $\{x_n:n\in \mathbb{N}\}$ in the unit ball of $Y$ such that $n \neq m$ implies ...
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2answers
293 views

compact projections to infinite dimensional Banach spaces

If I consider $X$ to be an infinite dimensional Banach space and $P\in P(X)$, that is, $P$ is a continuous linear projection. How does one prove that $P$ is compact if and only if $\dim R(P)$ is ...
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1answer
175 views

Is the range of this operator closed?

I think I am stuck with showing closedness of the range of a given operator. Given a sequence $(X_n)$ of closed subspaces of a Banach space $X$. Define $Y=(\oplus_n X_n)_{\ell_2}$ and set $T\colon ...
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61 views

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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162 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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157 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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240 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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189 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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2answers
677 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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486 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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487 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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148 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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569 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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185 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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70 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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141 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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266 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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130 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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424 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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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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842 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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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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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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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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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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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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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 ...
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314 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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420 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 ...