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Questions tagged [reflexive-space]

In functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space.

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Non-uniqueness of closest point in a closed, convex set

I have seen that if X is a Hilbert space and K is some non-empty, closed, convex subset then for every $x \in X $ there exists a $ unique \ y \in K $ which is closed to any other point of K. However, ...
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Showing that $A(\overline{B_1^X}) \subseteq Y$ is closed when $A \in \mathcal{L}(X,Y)$.

Exercise : Let $X,Y$ be Banach spaces and $X$ be reflexive, $A \in \mathcal{L}(X,Y)$. If $\overline{B_1^X} = \{ u \in X : \|u\|_X \leq 1\}$, show that $A(\overline{B_1^X}) \subseteq Y$ is closed. ...
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prove an operator is compact in reflexive space

In Banach spaces $E,F$, a compact operator $T\in \mathcal{L}(E,F)$ maps weakly convergent sequences into strongly convergent sequences. If E is reflexive, the converse is true. I need help in proving ...
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Is $X$ reflexive if $X=\cup Y_i $ and $Y_i$ is reflexive for every $i$?

Let $X$ be a Banach space. Assume that there are a family of closed (with respect to $\left\|\cdot\right\|_X$) subspaces $Y_i$ of $X$ such $X=\cup Y_i $ and $Y_i$ is reflexive for every $i$. Can we ...
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Why are reflexive spaces called like that?

A normed spaces $(X, \| \cdot \|)$ is called reflexive if the evaluation map $X \to X^{**}$ is an isomorphism. If $X$ is reflexive, it's not analytically distinguishable from it's bidual space $X^{**}$...
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Is every Banach space embedded in a reflexive space?

Can every Banach space be continuously embedded in a reflexive Banach space? As for example for $ L^\infty(\Omega)$ and $L^2(\Omega)$, where $ \Omega\subset R^n$ is a bounded open set.
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$A(D) \subseteq Y$ is closed, if $X$ is reflexive, $Y$ is Banach and $D \subseteq X$ is closed, convex and bounded.

Exercise : Let $X$ be a reflexive Banach space and $Y$ a Banach space. Also, let $A \in \mathcal{L}(X,Y)$ and $D \subseteq X$ be a closed, convex and bounded space. Show that $A(D) \subseteq Y$ is ...
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Question about notation in Banach spaces problem

Suppose that $X$ and $Y$ are Banach spaces, and that $T_X : X \to Y^*$ and $T_Y : Y \to X^*$ are both isometric isomorphisms. Show that if $(T_X x, y) = (x, T_Y y)$ for all $x \in X, y \in Y$, then $...
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Unit closed Ball in completion

Let $(E, \|\cdot\|_E)$ be any normed vector space. Consider the canonical identification with its double dual $(E'', \|\cdot\|_{E''})$ given by $$i_E: (E, \|\cdot\|_E) \to (E'', \|\cdot\|_{E''}), \; \...
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Examples reflexive spaces

Let $(E, \| \cdot \|_E)$ be a normed vector space over a field $\mathbb{K}$ and $(E', \| \cdot \|_{\mathrm{op}})$ its dual. Theorem 1). If $(E, \| \cdot \|_E)$ is reflexive, then each bounded ...
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$C(X)$ not reflexive if $X$ has infinitely many points.

Let $X$ be a compact metric space with infinitely many points, then show $C(X)$ is not reflexive. I think I see why this is the case for $X=[a,b]$, but I don't see how one can extend it. Is there a ...
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Non-reflexive space that is isomorphic to its second dual space

I was wondering if it is possible to construct a space that is non-reflexive (so it is not isomorphic to its second dual space under the cannonical embedding), but some isomorphism exists between them....
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Characterization of Reflexive Subspaces of finite signed-measures

I'm looking for a characterization of the reflexive subspaces of the space of finite measures on a measurable space $(\Omega,\mathcal{F})$. Maybe something, more informative than the unit ball is ...
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surjectivity of dual operator congruence implies reflexivity

I tried to prove the following: Let $X,Y$ be normed spaces and $\Phi:\mathcal{L}(X,Y)\to\mathcal{L}(Y^*,X^*)$, $\Phi(A)=A^*$ is surjective. Then $Y$ is reflexive. Here $X^*$ $(A^*)$ denotes the dual ...
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Point-wise convergence but not weakly

Let $X$ be a reflexive Banach space and $x_n\in C(0,\tau;X)$ be a bounded sequence. We know that a subsequence of $x_n$, denote it by the same symbol, converges weakly to $x$ in $L^2(0,\tau;X)$. Can ...
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infinite dimensional reflexive subspace of $C[0,1]$

We know that $C[0,1]$, the space of functions continuous on interval $[0,1]$ equipped with maximum norm is not reflexive. Is there any infinite dimensional reflexive subspace of $C[0,1]$ or every ...
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Is $C^1([a,b], \mathbb{R}^n)$ a reflexive Banach space?

I want to prove or disprove that $C^1([a,b], \mathbb{R}^n)$ equipped with the norm $||x||=\underset{t\in[a,b]}{\sup}|x(t)|_{\mathbb{R}^n}+\underset{t\in[a,b]}{\sup}|\dot{x}(t)|_{\mathbb{R}^n}$ is a ...
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Giving a proof to the weak compactness of the unity ball in a reflexive normed space

So, I am trying to prove that if $X$ is a normed and reflexive space, its unity ball, that is $B$=$\lbrace x\in X : \parallel x\parallel =1\rbrace$, is weakly compact. For that matter, I have already ...
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Question concerning specific part of proof that the dual of $L^p$ is $L^q$ where $p$ and $q$ are Hölder conjugates and $p<2$.

I want to show that if $p<2$ then the dual of $L^p([0,1])$ is $L^q([0,1])$ where $\frac{1}{p}+\frac{1}{q} = 1$ without referring to the fact that $L^p([0,1])$ is uniformly convex. My question ...
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Example: $F(t)$ is strongly continuous on $X$ but $F^*(t)$ is not

Let $X$ be a reflexive Banach space. I am looking for examples of the strongly continuous operators $F(t):t\mapsto \mathcal{L}(X)$ such their adjoint on $X$ for every $t$ is not a strongly continuous ...
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On reflexivity properties of the Schwartz space

Consider $\mathscr{S}$ the Schwartz space of rapidly decreasing complex smooth functions over $\mathbb{R}^{d}$, equipped with its usual metric topology, and $\mathscr{S}'$ its topological dual (the ...
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Every sequence has a weakly convergent subsequence with limit

Let $\mathbf{E}$ be a reflexive space and $\mathbf{A ⊂ E}$ be bounded and weakly closed. Show that $\mathbf{A}$ is sequentially compact, i.e., every sequence in $\mathbf{A}$ has a weakly convergent ...
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Compactness criterion for operator between reflexive Banach spaces

I found (without any proof) the following proposition: Let $T \in \mathcal{L}(X,Y)$ be a linear continuous operator between two reflexive Banach spaces $X,Y$, then $T$ is compact if and only if for ...
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Bounded sequence has Cauchy subsequence w.r.t. $ (x|y)_0:=\sum^\infty_{n=1} 2^{-n}\phi_n(x)\phi_n(y).$

Let $X$ be a separable reflexive real Banach space and let $(\phi_n)$ be a dense sequence in $$\{ \phi\in X'\,|\,\|\phi\|\leq 1 \}.$$ Consider in $X$ the scalar product $( \cdot| \cdot)_0 $ defined by ...
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How to show a normed space $X$ is reflexive if the weak topology and weak* topology on $X^*$ are equal?

Let $X$ be a normed space such that the weak topology and weak* topology on $X^*$ are the same. I want to show $X$ is reflexive. My attempt is: Since we can use $\sigma(X^*,\widehat{X})$ to denote ...
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A direct example on $X^*$ is reflexive but the weak topology and weak* topology on $X^*$ are not equal for a normed space $X$.

I asked this question Example on $X^*$ is reflexive but $\sigma(X^*,X)\neq\sigma(X^*,X^{**})$ for a normed space $X$. before. And the answer is to use $X$ is reflexive iff $\sigma(X^*,X)=\sigma(X^*,X^{...
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Example on $X^*$ is reflexive but $\sigma(X^*,X)\neq\sigma(X^*,X^{**})$ for a normed space $X$.

I just learned this theorem: Theorem: Let $X$ be a Banach space. Then the following statements are equivalent. (a) $X$ is reflexive. (b) $X^*$ is reflexive. (c) $\sigma(X^*,X)=\sigma(X^*,X^{**})$. ...
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How to get $\ell_1$ is not reflexive from $c_0^*$ is not reflexive?

I want to show $\ell_1$ is not reflexive. And I have already shown that $c_0^*$ is not reflexive. And I know there is an isometric isomorphism between $\ell_1$ and $c_0^*$. How to use the isometric ...
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Is the Unit ball of $X^{**} $ weak*- sequentially compact?

Am in the middle of a problem and i have the following conditions : Let $X$ be a reflexive Banach space with Schauder basis $(e_n)_{n=1}^{\infty}$, i have a sequence $x_n^{**} \in B_{X^{**}}\biggl(...
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My attempt to show the canonical embedding $c_0\rightarrow c_0^{**}$ is not surjective?

I want to show $c_0$ is not reflexive by showing the canonical embedding $c_0\rightarrow c_0^{**}$ is not surjective. I saw a reference said: since the inclusion map $c_0\rightarrow\ell_\infty$ is ...
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Non-Reflexivity of $L^1$ by finding a particular operator in the dual space?

Figured I might be able to come up with an interesting way to show that $L^1$ is non reflexive by finding an operator in the dual space that isn't bounded in the unit ball in $L^1$ and using some ...
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Weak and Weak* convergences implying reflexivity

Let $X$ be a Banach space. Suppose that for any sequence of functionals $(\phi_n) \subseteq X^*$ we have that $\phi_n$ converges weakly to some $\phi \in X^*$ if and only if $\phi_n$ converges weakly* ...
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Weak convergence in a reflexive, separable and infinite Banach space.

Let $Y$ be a infinite Banach space and $X$ be a reflexive, separable and infinite Banach space. Further let $T ∈\mathcal{L}(X,Y )$ satisies that $$\|Tx_n − Tx\|_{Y} \to 0,~~~~~as~~n\to\infty$$ , as $n →...
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Prove $L^p$ is reflexive for $1<p<\infty$ by using Riesz representation theorem

By definition, $X$ is reflexive if canonical injection $J:E \to E^{**}$ is surjective, where $\langle Jx,f\rangle_{E^{**},E^*}=\langle f,x\rangle_{E^*,E},~\forall x \in E,~\forall f \in E^*.$ In ...
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For $f$ in dual space, there exists $x$ with norm 1 and $f(x)=\|f\|$ if space is reflexive (and nontrivial)

Let $X\ne\{0\}$ be a reflexive space and let $f\in X^*$, where $X^*$ is the dual of $X$. I want to know: in general, does there exist an $x\in X$ with $\|x\|=1$, and $f(x)=\|f\|$, where $\|f\|$ is ...
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Best approximation in a reflexive normed space

Let $E$ be a reflexive normed space and let $\emptyset\neq K\subseteq E$ closed and convex. Show that then exists for every $x\in E$ a "best approximation" in $K$. Therefore a $y\in K$ with $\|x-y\|=d(...
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Reflexivity of intersection of two $L^p$ space

Q:Prove that the linear space $L^{p_1}\cap L^{p_2}$ $(p_1\neq p_2)$ with norm defined by $$\|f\|_{L^{p_1}\cap L^{p_2}}=\|f\|_{L^{p_1}}+\|f\|_{L^{p_2}}$$ is a reflexive Banach space. I've tried ...
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Reflexive but not separable space

I'm trying to find an example of normed vector space that is reflexive but not separable. (Separable but not reflexive is easy, for example $L^1$).