L-2 Space forms Hilbert Space

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Theorem

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

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

Let $\innerprod \cdot \cdot$ be the inner product on $\map {L^2} \mu$.


Then $\map {L^2} \mu$ endowed with $\innerprod \cdot \cdot$ is a Hilbert space.


Proof



Sources