Riesz Representation Theorem (Hilbert Spaces)/Examples/L2 Space
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Example 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:
- $\ds \forall f \in \map {L^2} \mu: \map F f = \int f \overline{f_0} \rd \mu$
Proof
By $L^2$ Space forms Hilbert Space, $\map {L^2} \mu$ is a Hilbert space with the $L^2$ inner product $\innerprod \cdot \cdot$.
Hence, the Riesz Representation Theorem (Hilbert Spaces) applies, so that there exists a unique $f_0 \in \map {L^2} \mu$ such that:
- $\forall f \in \map {L^2} \mu: \map F f = \innerprod f {f_0}$
Unpacking the definition of the $L^2$ inner product, the result follows.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem: Corollary $3.5$