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Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator.

a) Let $x\in [a,b]$. Show that there is a unique $g_{x}\in H$ with $\langle f,g_x\rangle=(Tf)(x)$ and all $f \in H$.

My solution:

Let $x\in[a,b]$, define the linear continuous map $L_{x}:H\rightarrow \mathbb{K}$, with $\mathbb{K}=\mathbb{C}$ or $\mathbb{K}=\mathbb{R}$ by $$ L_{x}(f):=T(f)(x).$$

Since $H$ is an Hilbert space and $L_{x}:H\rightarrow \mathbb{K}$ is a bounded linear functional on $H$ we can apply the Riesz-Frechet theorem. According to the Riesz-Frechet theorem there exists a unique $g_{x}\in H$ such that for $x\in[a,b]$ and all $f\in H$ $$L_{x}(f)=T(f)(x)=(Tf)(x)=<f,g_{x}>.$$

Question 1: Is this correct? Or am I missing something?

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  • What is the exact statement of the Riesz Representation Theorem? –  Mar 26 '13 at 10:04
  • The one I have in my lecture notes is as folllows: Let $H$ be a Hilbert space $\phi:H\rightarrow\mathbb{K}$ be a bounded linear functional on $H$. Then there exists a unique $g\in H$ such that $\phi=\phi_{g},$ i.e., $\phi(f)=<f,g>$ for all $f\in H$. – Lech121 Mar 26 '13 at 10:07

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