---- Let $\mathbb{F}$ [[field|denote]] $\mathbb{R}$ or $\mathbb{C}$. > [!definition] > Let $(X, \Sigma, \mu)$ be a [[measure|measure space]] and $1 \leq p \leq \infty$. There is a natural [[continuous]][^6] [[bilinear map|bilinear form]] $\langle -,- \rangle$ [[Lp-norm|on]] $L^{p'} \times L^{p}$ [[dual exponent|given by]] $\langle f,g \rangle = \int f g \, d\mu ,$ which we may call the **Hölder pairing** for $(p,p')$.[^4] ^definition [^4]: Sometimes $g$ is [[conjugate|conjugated]] in this definition, making the pairing sesquilinear instead of bilinear over $\mathbb{C}$, but otherwise not changing any downstream results. [^6]: Continuity of $\langle f,g \rangle$ follows straight from [[Hölder's inequality]]: $|\langle f,g \rangle| \leq \|f\|_{p} \|g\|_{p'}$, meaning $\langle -,- \rangle$ is [[operator norm|bounded]] (operator norm=1). > [!theorem] Theorem. ([[Lp duality]]) > - ($1<p<\infty$) The Hölder pairing $\langle -,- \rangle: L^{p'} \times L^{p} \to \mathbb{F}$ is [[perfect pairing|perfect]] in the [[Banach space|Banach]] [[category]] for $1 < p < \infty$: [[dual vector space|the natural maps]] $\begin{align} L^{p'} &\to (L^{p})^{\vee} ,& L^{p} \to (L^{p'})^{\vee} \\ f & \mapsto \langle f, - \rangle & g \mapsto \langle -, g \rangle \end{align}$ > are [[linear isomorphism|linear]] [[homeomorphism|homeomorphisms]] for $1<p<\infty$. > > > - $(p=1,p'=\infty)$ The natural map $\begin{align} L^{\infty} &\to (L^{1})^{\vee} \\ f & \mapsto \langle f, - \rangle \end{align}$ >is a [[linear isomorphism|linear]] [[homeomorphism]] as well if we additionally assume $\mu$ is a [[finite measure|σ-finite]] [[measure]]. > >- ($p=\infty, p'=1$) The natural map $\begin{align} L^{1} &\to (L^{\infty})^{\vee} \\ f & \mapsto \langle f, - \rangle \end{align}$ is always a [[linear map|linear]] [[topological embedding]], but in general fails to be [[surjection|surjective]]. > Moreover, in all cases the witnessing [[isomorphism]] (or [[topological embedding|embedding]]) is an [[Lipschitz continuous|isometry]] of [[Banach space|Banach spaces]]: $\|\langle f,- \rangle\|=\|f\|_{p'}$ for all $f \in L^{p'}$. > ^theorem > [!equivalence] > Phrasing in terms of the [[inverse map]], the result is saying that there exists an isometric isomorphism $\pi:(L^{p})^{\vee} \to L^{p'}$ characterized by[^5] the property that, for all $\varphi \in (L^{p})^{\vee}$, $g \in L^{p}$, $\varphi g= \langle \pi(\varphi), g \rangle = \int \pi(\varphi)g \, d\mu .$ ^equivalence [^5]: Explicitly: we start with [[Lp duality|Hölder pairing]] $\langle -,- \rangle:L^{p'} \times L^{p} \to \mathbb{F}$, inducing an isomorphism $((f \in L^{p'}) \xmapsto{\Psi} (\langle f,- \rangle \in L^{p})^{\vee})$ whose inverse we've defined to be $\pi:(L^{p})^{\vee} \to L^{p'}$. Applying $\pi$ to $\varphi \in (L^{p})^{\vee}$, followed by by $\Psi=\pi ^{-1}$, we get $\varphi(g \in L^{p})=\big((\Psi \circ \pi)(\varphi)\big)(g )= \langle \pi \varphi, g \rangle .$ > [!note] Remarks. > - Proving [[injection|injectivity]] and [[Lipschitz continuous|isometricity]] is a straightforward application of [[Hölder's inequality]] (and [[converse to Hölder's inequality|its converse]]). Proving [[surjection|surjectivity]] is not easy, and makes use of results such as the [[Radon-Nikodym Theorem]]. > - Typically $(L^{\infty})^{\vee} \supsetneq L^{1}$, but sometimes the image onto which the isometric embedding $L^{1} \hookrightarrow (L^{\infty}) ^{\vee}$, $g \mapsto \langle -, g \rangle$, surjects has nice known structure, e.g. see [[the dual of the space of null sequences naturally identifies with l1]]. ^note > [!proof]- Proof. ([[Lp duality]]) > Omitted in our course. ---- #### ----- #### 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 > ```