Riesz-Fischer theorem

For

1 \leq p \leq \infty

, the Lebesgue space

L^{p}

is complete in its usual norm.

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L2 space is a Hilbert space
Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 1 / iii / Solution
Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 22H / a / Solution

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Riesz-Fischer theorem by

Ciro Santilli  40 Updated 2025-07-16

A measurable function defined on a closed interval is square integrable (and therefore in

L^{2}

) if and only if Fourier series converges in

L^{2}

norm the function:

lim_{N \to \infty} ∥ S_{N} f - f ∥_{2} = 0

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