---- > [!definition] Definition. ([[reflexive space]]) >Let $X$ be a [[Hausdorff space|Hausdorff]] [[locally convex space|locally convex]] [[topological vector space|topological]] [[vector space|vector]] [[topological space|space]]. The [[natural transformation|natural]] [[linear map|map]] into the [[double dual of a finite-dimensional vector space is naturally isomorphic to that space|double dual]][^1] $\begin{align} X &\to X^{ \vee \vee} \\ x& \mapsto \_(x) \end{align}$ is always a [[linear map|linear]] [[continuous]] [[injection]] (usually in fact a [[topological embedding]], e.g. if $X$ is [[Banach space|Banach]], if $X$ banach is moreover [[Lipschitz continuous|isometry]], clean this up). When it is in fact a [[homeomorphism]], we call $X$ **reflexive**. (For [[Banach space|Banach spaces]] we only require [[surjection|surjectivity]] in light of the [[open mapping theorem]].) ^definition - [ ] what is the topology on the double dual? will discuss later (but probably not in this note) > [!equivalence] Equivalence for Normed Vector Spaces. > - [[weak ball characterization of reflexive normed spaces]] ^equivalence > [!basicexample] > - Any [[dimension|finite-dimensional]] [[norm|normed]] [[vector space]] is reflexive; [[double dual of a finite-dimensional vector space is naturally isomorphic to that space|see here]] >- [[Lp-norm|For]] $1<p<\infty$, $L^{p}(E)$ is reflexive for any [[measurable function|measurable]] $E \subset \mathbb{F}^{n}$, as follows from [[Lp duality]].[^2] In turn, the [[Sobolev space]] $W^{k,p}(E)$ is reflexive for $1<p<\infty$. ^basic-example [^1]: This notation is $\Phi_{f}=\_(f): X^{\vee} \to \mathbb{F}$ given by the evaluation of the argument $\_$ at $f$, that is $\Phi_{f}(\varphi: X \to \mathbb{F})=\varphi(f)$. [^2]: Verifying this is bookkeeping. Since $p,p' \in (1, \infty)$, we have canonical isomorphisms $\Psi:L^{p} \to (L^{p'})^{\vee}$, $\Psi(f)=\langle f, - \rangle$, and $\Xi:L^{p'} \to (L^{p})^{\vee}$, $\Xi(g)=\langle -,g \rangle$. Write $\pi:=\Xi ^{-1}:(L^{p})^{\vee} \to L^{p'}$, and denote by $\pi^{\vee}$ its [[dual map]] $\begin{align} \pi^{\vee}: (L^{p'})^{\vee}& \to (L^{p})^{\vee \vee } \\ (\varphi: L^{p'} \to \mathbb{F}) & \mapsto \big( \varphi \circ \pi: (L^{p})^{\vee} \to \mathbb{F} \big). \end{align}$ This gives us a map $L^{p} \xrightarrow{\Psi} (L^{p'})^{\vee} \xrightarrow{\pi^{\vee}} (L^{p})^{\vee \vee}.$ We claim $\pi^{\vee} \circ \Psi$ equals the map $f \mapsto \_(f)$. Indeed, let $f \in L^{p}$ and let $\varphi \in (L^{p})^{\vee}$. Then $(\pi^{\vee} \Psi)(f)(\varphi)=\varphi(f)$, as one has $\begin{align} (\pi^{\vee} \Psi)(f) \ (\varphi) &= \pi^{\vee} (\langle f, - \rangle ) \ (\varphi) \\ &= (\langle f, \pi(-) \rangle ) \ (\varphi) \\ &= \langle f, \pi(\varphi) \rangle \\ &= \varphi(f), \end{align}$ where the last step uses that $\pi(\varphi)$ is the unique element $g \in L^{p'}$ satisfying $\langle h, g \rangle=\varphi(h)$ for all $h \in L^{p}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```