Proof of Riesz Representation

Riesz representation is a theorem stating that in a Hilbert space \(\mathcal{F}\), every continuous linear functional \(L:\mathcal{F}\rightarrow\mathbb{R}\) can be written as \(Lf=\left\langle f,g\right\rangle _{\mathcal{F}}\) for a unique \(g\in\mathcal{F}\).

Proof. Assume that the linear functional \(L\) is continuous. For the edge case where \(Lf=0\) for all \(f\), this is trivial because \(Lf=\left\langle f,0\right\rangle \), so \(g=0\). If not, we know that the null space of \(L\) i.e., \(N=\{f\mid Lf=0\}=L^{-1}\left(\{0\}\right)\) is closed because a continuous operator maps a closed set to a closed set i.e., \(\{0\}\). Since \(N\) is closed, the orthogonal complement \(N^{\perp}\) (subspace of elements orthogonal to \(N\)) contains elements other than \(0\) element. Consider \(h\in N^{\perp}\) such that \(\|h\|_{\mathcal{F}}=1\). Consider \(u_{f}=\left(Lf\right)h-\left(Lh\right)f\in N\) (i.e., \(Lu_{f}=0\)). Since \(h\) and \(u_{f}\) are orthogonal by construction, we have \[\begin{eqnarray*} 0 & = & \left\langle u_{f},h\right\rangle _{\mathcal{F}}\\ & = & \left\langle \left(Lf\right)h-\left(Lh\right)f,h\right\rangle _{\mathcal{F}}\\ & = & \left(Lf\right)\|h\|^{2}-\left(Lh\right)\left\langle f,h\right\rangle \\ & = & Lf-\left\langle f,(Lh)h\right\rangle _{\mathcal{F}}\\ \Rightarrow Lf & = & \left\langle f,(Lh)h\right\rangle _{\mathcal{F}}. \end{eqnarray*}\] Hence, \(g=\left(Lh\right)h\) for some \(h\in N^{\perp}\).

The element \(g\) is unique. If there were \(g_{1}\) and \(g_{2}\), then \(0=|Lf-Lf|=|\left\langle f,g_{1}-g_{2}\right\rangle |\leq\|f\|\|g_{1}-g_{2}\|\) by Cauchy-Schwarz inequality. This implies \(\|g_{1}-g_{2}\|=0\). Hence \(g_{1}=g_{2}\). Riesz representation is useful in establishing the existence of a mean embedding in RKHS of a probability distribution.

Reference: Dino Sejdinovic's teaching slides