Riesz Representation Theorem (Hilbert Spaces)/Examples/Space of Square Summable Mappings

Example of Use of Riesz Representation Theorem (Hilbert Spaces)
Let $\map {\ell^2} \N$ be the space of square summable mappings on $\N$.

Let $N \in \N$.

Let $L_N: \map {\ell^2} \N \to \GF$ be defined by:


 * $\map {L_N} {\sequence{ a_n } } := a_N$

Let $\delta_N \in \map {\ell^2} \N$ be given by:


 * $\forall n \in \N: \paren{ \delta_N }_n = \begin{cases}

1 & n = N \\ 0 & n \ne N \end{cases}$

Then for all $a \in \map {\ell^2} \N$:


 * $\map {L_N} a = \innerprod a {\delta_N}$

Proof
By Space of Square Summable Mappings is Hilbert Space, $\map {\ell^2} \N$ is a Hilbert space.

Since for all $a \in \map {\ell^2} \N$:

it follows that $L_N$ is a bounded linear functional.

Hence the Riesz Representation Theorem (Hilbert Spaces) applies, so that there exists a unique $b \in \map {\ell^2} \N$ such that for all $a \in \map {\ell^2} \N$:


 * $\map {L_N} a = \innerprod a b$

Let us check that $\delta_N$ fulfils the claim:

The result follows.