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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.

In the area of mathematics known as 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. Reflexive Banach spaces are often characterized by their geometric properties.

Let $X$ be a normed space and $X^{\ast \ast}$ denote the second dual vector space of $X$. The canonical map $x \mapsto \hat{x}$ defined by $\hat{x}(f) = f(x), f\in X^{*}$ gives an isometric linear isomorphism (embedding) from $X$ into $X^{**}$. The space $X$ is called reflexive if this map is surjective.

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A Banach space is reflexive if its dual is reflexive (and conversely)

How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?
Ken
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Prove that every reflexive Banach space is weakly (sequentially) complete.

Here is the question: A sequence $(x_{n})$ in a normed linear space $X$ is weakly Cauchy if $(Tx_{n})$ is a Cauchy sequence for every $T \in X^*.$ The space $X$ is weakly (sequentially) complete if every weakly Cauchy sequence in $X$ is weakly…
user591668
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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 every sequence $\left(x_n\right)_{n\in\mathbb{N}}…
Enrico Polesel
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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* to $\phi.$ We know that the weak and weak*…
N. Geis
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Proving that the quotient $\ell^\infty/c_0$ is not reflexive

Let $\ell^\infty$ be the space of bounded sequences with the maximum norm and $c_0$ the space of sequences that have limit $0$ with the same norm. I use the notation $E^*$ for the dual of a normed space $E$. I know that $c_0$ is a closed subspace of…
Mipeal
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Covering a set with $N$ balls of common minimal(!) radius (Existence of such a covering)

Let $X$ be a Banach space, and denote by $B_r (x)$ the closed ball of radius $r > 0$ around $x \in X$. Furthermore, let $A \subset X$ be compact and $N \in \Bbb{N}$. I am interested in "optimally" covering $A$ by $N$ balls, i.e,. with as small…
PhoemueX
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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 subsequence with limit in $\mathbf{A}$. This…
user574356
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How many points does a line intersect a sphere in an infinite-dimensional normed vector space?

Let $(E, |\cdot|)$ be a n.v.s. We fix $r>0$ and $x,y \in B(0, r)$ such that $x\neq y$. Here $B(0, r)$ is the open ball centered at the origin and having radius $r$. The set of all points in the line though $x$ and $y$ is $$ \{tx+(1-t)y \mid t \in…
Akira
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Suppose $X$ is a Banach space and $Y$ is a closed subspace. If $Y$ and $X{/}Y$ are reflexive, then is $X$ also reflexive?

My intuition is that the answer is affirmative. I have shown that a Banach space $X$ is reflexive iff the closed unit ball of $X$ is weakly compact. I was trying to use this, but it hasn't been a success so far.
MathWiz
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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 space of tempered distributions). I have a few…
CarrizoV
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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 defined as $\sup\{|f(x)|:x\in X,\|x\|=1\}$? I know…
user316769
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Proof that every finite-dimensional normed space is reflexive

Let $X$ be a finite-dimensional normed vector space. Let $X^*$ denote the space of linear dual space of $X$, i.e. $X^*=L(X,\mathbb{C})$ and let $X^{**}$ be the dual space of $X^*$. For each $x\in X$, define $\hat{x}:X^*\to \mathbb{C}$ by…
stph
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Prove that a Banach space cannot be reflexive if some strict closed subspace of its dual space separates its points

Let $X$ be a Banach space and let $Z$ be a closed subspace of $X^*$ such that $Z\neq X^*$. Suppose $Z$ separates the points in $X$, that is, if $x \in X$ and $x^*(x) = 0$ $\forall x^* \in Z$ then $x = 0$. Prove that $X$ is not reflexive, i.e. that…
14Lucas07
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James's theorem in incomplete normed spaces

One version of James's theorem in functional analysis states the following. A Banach space is reflexive if, and only if, every bounded linear functional attains its norm on the unit sphere. For example if $X$ is a reflexive Banach space, then by…
SmileyCraft
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