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
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Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples