Riesz Representation Theorem (Hilbert Spaces)/Examples

Examples of Use of Riesz Representation Theorem (Hilbert Spaces)

Let $\struct{ X, \Sigma, \mu }$ be a measure space.

Let $\map {L^2} \mu$ be the associated $L^2$ space.

Let $F: \map {L^2} \mu \to \GF$ be a bounded linear functional.

Then there exists a unique $f_0 \in \map {L^2} \mu$ such that:

$\ds \forall f \in \map {L^2} \mu: \map F f = \int f \overline{f_0} \rd \mu$

Let $\map {\ell^2} \N$ be the space of square summable mappings on $\N$.

Let $N \in \N$.

Let $\GF$ be a subfield of $\C$.

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:

$\forall a \in \map {\ell^2} \N: \map {L_N} a = \innerprod a {\delta_N}$

where $\innerprod \cdot \cdot: \map {\ell^2} \N \times \map {\ell^2} \N \to \GF$ denotes the inner product on $\map {\ell^2} \N$.