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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Does there exist a linear map from $\ell^1$ to its bi-dual which is isometrically isomorphic?

I know that the space $\ell^1$ is not reflexive as the dual space $\ell^\infty$ is not separable but I don't know the dual space of $\ell^\infty$. Someone, please provide me with the details of the ...
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Tensor product of linear forms.

Let $P$ be a module over a conmutative unitary ring $A$ and $\mathcal{J}^r(P)$ be the module of $r$-linear forms $P \times \cdots \times P \longrightarrow A.$ If $\alpha \in \mathcal{J}^r(P)$ and $\...
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Necessary assumptions for direct method of calculus of variations

In "Calculus of Variations" by F. Rindler I learned about the following result (direct method of calculus of variations): If $X$ is a topological space, $f: X \to \mathbb{R}$ lower ...
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Prove that a Banach space cannot be reflexive if some strict closed subspace of its dual space separates its points [closed]

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 = ...
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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 ...
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Reflexivity and Separability of $L^p(0,T; X)$ spaces

I am familiar with the separability and reflexivity conditions for $L^p$ spaces. How do they generalize for Bouchner spaces $$L^p(0,T; X)$$ Do they follow from separability and reflexivity of $X$? ...
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How can a non-reflexive space have a reflexive subspace?

I know that this can happen (take any one-dimensional subspace, for instance), but I had the following thought while reading Kadison and Ringrose (Fundamentals of the Theory of Operator Algebras, Vol ...
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How does the reflexivity of $V$ imply the denseness of $R(i^*)$?

For a linear map $T$, we denote by $R(T)$ its range and by $N(T)$ its kernel. I'm reading about Gelfand triple at page 136 in Brezis' Functional Analysis. Let $(H, \langle \cdot , \cdot \rangle_H)$ ...
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Example of isometry between reflexive and non-reflexive Banach spaces

Let $E,F$ be real Banach spaces where $E$ is reflexive. Let $i:E \to F$ be a surjective isometry. If $i$ is linear, then $F$ is also reflexive. Could you give an example where $F$ is not reflexive? ...
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When weakly convergence implies the norm convergence?

Suppose I have a finite measure space $(X,\Sigma,\mu)$. I have doubts about the following statement: In a reflexive Banach space weakly compact set implies the norm compact. Is the following statement ...
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Reflexivity is invariant under isomorphism [closed]

Let $X$ and $Y$ be two normed space. We suppose that $T\colon X\to Y$ is an isomorphism. It's true or false that $$X\;\text{Reflexive}\implies Y\;\text{Reflexive}?$$ I would like to answer yes, but ...
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Showing that $\ell^p$ with $1<p<\infty$ is reflexive using the adjoint operator.

I have to show that for all $1<p<\infty$ the space $\ell^p$ is reflexive using the adjoint operator. Now, we say that the application $T\colon (\ell^p)'\to \ell^q$ defined as $$T(f(x))=(f(e_1),...
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Reflexivity of infinite dimensional vector spaces and existence of non degenerate bilinear forms

How can an infinite dimensional vector space like (most of) the Lebesgue or Sobolev spaces even be reflexive? If an infinite dimensional vector space cannot be isomorphic to its dual space, the how ...
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Brezis' exercise 6.2.1: If $E$ is reflexive, then $T(B_E)$ is closed in norm topology

I'm trying to solve an exercise in Brezis' Functional Analysis Let $E$ and $F$ be Banach spaces and $T:E \to F$ a bounded linear operator. Let $B_E$ be the closed unit ball of $E$. If $E$ is ...
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Is the sum of a closed unit ball and a closed set itself closed?

I am reading here that in a Banach space, the sum of the closed unit ball with a closed bounded convex set might fail to be closed itself. It seems there is a counterexample if and only if the ...
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Brezis' exercise 4.16.1

Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Let $p \in (1, \infty)$ and $p'$ its Hölder conjugate. Let ...
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Variational formulation how to find it

I have this problem $$ - \operatorname{div}(|\nabla u|^{p-2} \nabla u) =f(u), x\in \Omega $$ With $\Omega $ is bounded in $\mathbb{R}^N$ and $u=0$ on $\partial \Omega $ the condition on $f$ is $f\in W^...
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example of non-reflexive space with a subspace having trivial annihilator

I would like to find an example of a non-reflexive normed vector space $X$ and a proper closed subspace $W \subset X^*$ such that $ ^\perp W = \{0\}$. I was thinking maybe taking $X = C^0([0,1])$ the ...
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Minimization in reflexive Banach spaces

Let's consider the following problem. Let $A,C$ be two nonempty sets in a normed space $X$. Let $R(A;C)$ denote the set of real numbers $r \ge 0$ such that $A$ is contained in a closed ball of radius ...
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Dual spaces isomorphic

Let $X$ and $Y$ be Banach Spaces. I have to prove if $X'$ and $Y'$ are isomorphic and $X$ is reflexive, then $X$ and $Y$ are isomorphic. My idea. Since $X$ is reflexive we have that $X'$ is also ...
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Brezis Functional Analysis Exercise 3.14 (A Variation of Helly's Theorem)

I am stumped at showing exercise 3.14 on Brezis' Functional Analysis: Let $E$ be a reflexive Banach space and $I$ a set of indices. Consider a collection $(f_{i})_{i \in I}$ in the dual space $E'$ and ...
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Normed space constructed from the iterated double duals of a Banach space

If $X$ is a Banach space, and $X^{(n+1)} = (X^{(n)})^*$ where $X^{(0)} = X$, then the sequence $X^{(2n)}$ stabilizes iff $X$ is reflexive. Consider $\hat{X} = \bigcup_{n=0}^\infty X^{(2n)}$ where $X^{(...
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Extending a linear functional on a closed subspace

I was trying to prove the following statement "A closed subspace $Y$ of a reflexive Banach space $X$ is also reflexive." I followed the steps similar to this answer on math.stackexchange ...
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Do symmetric monoidal functors preserve reflexivity?

Suppose that $(C, \otimes_C, \mathcal{H}om_C, I_C), (D, \otimes_D, \mathcal{H}om_D, I_D)$ are closed symmetric monoidal categories. Let $$ F: (C, \otimes_C, I_C) \rightarrow (D, \otimes_D, I_D) $$ ...
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Reference request: If $E$ is a reflexive Banach space, then $L_p(X, \mu, E)$ is reflexive and $(L_p(X, \mu, E))^* = L_q(X, \mu, E^*)$

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p(X, \mu, E)$. I read somewhere that If $E$ is reflexive, then $L_p(...
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Let $E$ be a Hilbert space and $p \in (1, \infty)$. Then $L_p(X, \mu, E)$ is reflexive

I'm reading Brezis's Functional Analysis, Sobolev Spaces and Partial Differential Equations, and trying to generalize Theorem 4.10 from $\mathbb R$ to to a Hilbert space $E$. Could you have a check on ...
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Norm equivalence preserves canonical evaluation map and thus reflexivity

I'm trying to prove below result (that I'm not sure if it's true!). Could you have a check on my attempt? Theorem: Let $E = F$ be non-empty sets. Let $|\cdot|, [\cdot]$ be norms on $E, F$ ...
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Prove that $c_0$ is not reflexive

I read a simple proof of showing $c_0$ is not reflexive. It says that $c_0$ is separable and $l^\infty$ is not separable so the canonical map from $c_0$ to $c_0^{**}=l^\infty$ is not an isomorphism. ...
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Using category theory, prove that a Banach space is reflexive iff its dual is reflexive

I'm trying to prove the following result A banach space is reflexive if and only if its dual is reflexive in the most category theoretical way I can. I managed to prove one implication in a not too ...
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Dual of space of compactly supported distribution is space of smooth functions.

Let $(M,g)$ be the compact manifold with boundary. Let $\mathcal{E}^{\prime}(M)$ denotes the set of distributions (continuous linear functionals $)$ on $C^{\infty}(M)$, we equip this space with the ...
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The existence of a sequence in a non reflexive Banach space

Let $X$ be a Banach space, and $x\in X$. Is it true that there exists $(x^*_n)$ a sequence in $X^*$ such that $\|x^*_n\| \leq 1$ and $x^*_n(x)\xrightarrow[n \to \infty]{}\|x\|$? It's true if $X$ is ...
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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 ...
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Exercise 5.16 from Brezis' Functional Analysis

Suppose $H$ is a Hilbert space with scalar product $(\cdot , \cdot )$ and norm $| \cdot |$. Let $V \subseteq H$ be a dense linear subspace and say it has its own norm $\| \cdot \|$ such that $V$ is ...
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A Space is Reflexive if its Image under the Canonical Injection is Reflexive?

Consider the following corollary in Brezis: Here is part of the proof of it: It was mentioned that if $J(E) \subseteq E^{**}$ is reflexive, then $E$ is also reflexive, where $J: E \to E^{**}$ be the ...
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Reflexivity and Separability of $L^{\infty}$ and $L^{1}$

I am currently trying to understand the following proposition: Let $\mu$ be a Radon measure on $\mathbb{R}^n$. Then: (1) $L^{\infty}(\mu)$ is neither reflexive nor separable (2) $L^{1}(\mu)$ is not ...
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Qustion about attaining of the functional norm

According to James's theorem, a Banach space X is a reflexive space if and only if $$ \forall f \in X^* ~ \exists x (\| x \| = 1): |f(x)| = \| f \|. $$ It is known that $L_1 [0; 1]$ is not reflexive. ...
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Spectrum of positive linear operators defined for every $L^p$ space.

Consider the probability space $([0,1], \mathcal B([0,1]), \lambda(\mathrm{d} x) )$, where $\mathcal B([0,1])$ is the Borel $\sigma$-algebra and $\lambda$ the Lebesgue measgure on $[0,1]$. Let $P,T:L^...
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Intuition on why Reflexive Spaces are Important

Why are reflexive spaces important intuitively? I know that it brings about many good properties: Properties of reflexive Banach spaces. However, what is so special about reflexive spaces that make ...
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How are the metric structure and infiniteness of $K$ essential in Brezis's Ex 3.25?

I'm doing Ex 3.25 in Brezis's book of Functional Analysis. Let $K$ be a compact metric space that is not finite. Prove that $\mathcal C(K)$ is not reflexive. In below attempt, I only use the ...
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Brezis's Ex 3.22: If $E$ is reflexive, there is a sequence of norm $1$ that weakly converges to $0$

I'm doing Ex 3.22 in Brezis's book of Functional Analysis. Let $E$ be an infinite-dimensional Banach space satisfying one of the following assumptions: (a) $E'$ is separable (b) $E$ is reflexive. ...
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A proof without using net in Brezis's Ex 3.14

I'm doing Ex 3.14 in Brezis's book of Functional Analysis. Let $E$ be a reflexive Banach space and $I$ a set of indices. Consider a collection $(f_{i})_{i \in I}$ in the dual space $E'$ and a ...
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A simpler proof of Milman–Pettis theorem by diameter argument

Recently, I have come across an elegant proof of Milman–Pettis theorem. Surprisingly, I'm able to make this proof even simpler. I'm very happy to share it with you and receive your suggestion. Let $E$...
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An intuitive exposition of a proof of Milman–Pettis theorem

I've recently read the proof of Milman–Pettis theorem in Brezis's book of Functional Analysis. Let $E$ be a uniformly convex Banach space. Then $E$ is reflexive. I try to get some feeling of how the ...
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Where does my proof of Milman-Pettis theorem break down?

I'm trying to prove Milman-Pettis's theorem. Let $E$ be a uniformly convex Banach space. Then $E$ is reflexive. Clearly, my attempt is not correct because I have not used the uniform convexity of $E$...
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Reflexive Banach space. [duplicate]

Let $X$ be a Banach space. Then we say that $X$ is reflexive if the map $J:X\to X^{**}$ defined by $J(x)(f):=f(x)$ is an isometric isomorphism. Here $X^{**}$ is the double dual of $X$ and $f\in X^{*}$ ...
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Let $E$ be a reflexive Banach space and $M$ its closed linear subspace. Then $M$ is reflexive

I'm trying to re-phrase this proof in Brezis's book of Functional Analysis. Let $E$ be a reflexive Banach space and $M$ its closed linear subspace. Then $M$ is reflexive. Could you have a check if I ...
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If $E,F$ are reflexive Banach spaces, then the graphs of unbounded linear operators $A$ and $A''$ are isometrically isomorphic

I'm reading Theorem 3.24 in Brezis's book of Functional Analysis. The statement of the theorem is: Let $E$ and $F$ be two reflexive Banach spaces. Let $ A:D(A) \subseteq E \rightarrow F$ be an ...
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Do Kakutani's, Helly's, and Goldstine's theorems all hold for general normed linear spaces?

I'm reading the proof of Kakutani's theorem in Brezis's book of Functional Analysis. Theorem 3.17 (Kakutani). Let $E$ be a Banach space. Then $E$ is reflexive if and only if $$B_{E}=\{x \in E ;\|x\| \...
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If $(^0E)^0=E$ for every closed subspace $E\subset X^*$ then $X$ is reflexive.

Let $X$ be a normed vector space, $X^*$ be its dual and $X^{**}$ be its bidual. For every $E \subset X^*$ we define the left-annihilator of $E$ as $$^0E=\{x \in X : f(x)=0 \ \forall f \in E\}.$$ And ...
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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 ...
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