---- [[field|For now]], assume $\mathbb{F}=\mathbb{R}$. Obviously the case of $\mathbb{C}$ is important but our course really only cares about $\mathbb{R}$ at the moment. > [!theorem] Theorem. ([[Riesz Representation Theorem]]) > If $V$ is an $\mathbb{R}$-[[Hilbert space]], then the [[inner product]] $\langle -,- \rangle$ is a [[perfect pairing]] with respect to the ([[continuous]]) [[dual vector space|dual space]] $V^{\vee}$. Moreover, the[[musical isomorphism induced by a nondegenerate bilinear form|]] [[musical isomorphism induced by a nondegenerate bilinear form|pertinent]] [[linear map|linear]] [[homeomorphism|homeomorphism]] $ \begin{align} V & \xrightarrow{\flat} V^{\vee} \\ v & \mapsto \langle v, - \rangle \end{align}$ is [[natural transformation|natural]] and is an [[Lipschitz continuous|isometry]]. ^theorem - [ ] version over $\mathbb{C}$ - [ ] specific form of $\flat$ in terms of orthonormal basis > [!specialization] > (Assuming the base field is $\mathbb{R}$ for brevity.) Let $(X, \Sigma, \mu)$ be a nontrivial [[measure|measure space]]. [[Lp-norm|Consider]] the [[vector space|space]] $L^{p}(\mu)$, $1<p<\infty$. In this case, [[Lp duality]] says the [[Lp duality|Hölder pairing]] $\langle f,g \rangle= \int f {g} \, d\mu$ is [[Banach space|Banach]]-[[perfect pairing|perfect]]. Recall that if (and only if) $p=2$, then $X$ [[nontrivial Lp spaces are Hilbert iff p=2|is in fact a]] [[Hilbert space]] under $\langle -,- \rangle$. > Thus, while the two results in general apply to different objects, the statements of RRT and $L^{p}$-duality coincide for $L^{2}(\mu)$. ^specialization > [!proof]- Proof. ([[Riesz Representation Theorem]]) > ~ ---- #### > [!specialization] Statement and proof for [[dimension|finite dimensions]]. Suppose $V$ is an [[inner product space]] that is finite-dimensional and $\varphi$ is a [[linear functional]] on $V$. Then there is a unique [[vector]] $u \in V$ such that $\varphi(v) = \langle v,u \rangle$ for every $v \in V$. Specifically, if $\{ e_{j} \}_{j=1}^{n}$ is an [[orthonormal basis]] of $V$ we have $u=\overline{\varphi(e_{1})}e_{1}+ \dots + \overline{\varphi(e_{n})}e_{n}.$ > > > > [!proof]- Proof. > > First we show existence. . [[writing a vector as a linear combination of orthonormal basis using projections|We have]] $\begin{align} > > \varphi(v)= & \varphi(\langle v,e_{1} \rangle e_{1} + \dots + \langle v,e_{n} \rangle e_{n} ) \\ > > = & \langle v,e_{1} \rangle\varphi(e_{1}) + \dots + \langle v,e_{n} \rangle \varphi(e_{n}) \\ > > = & \langle v, \overline{\varphi(e_{1})} e_{1} + \dots + \overline{\varphi(e_{1})} e_{n} \rangle > > \end{align}$ > > for every $v \in V$. > > Thus setting $u= \overline{\varphi(e_{1})} e_{1} + \dots + \overline{\varphi(e_{1})} e_{n}$ gives us existence. > > > > Next we show uniqueness. Suppose $u_{1},u_{2} \in V$ such that $\varphi(v)=\langle v,u_{1} \rangle \and \varphi(v)=\langle v,u_{2} \rangle $ > > for every $v \in V$. > > This implies $\langle v,u_{1} \rangle = \langle v,u_{2} \rangle$, hence $0=\langle v,u_{1}-u_{2} \rangle$. Since this holds for every $v \in V$, it in particular holds for $v=u_{1}-u_{2}$. Taking this we obtain $u_{1}=u_{2}$. $\qedin$ > > > > > ----- #### 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 > ```