$𝐿 2$ space

Let $(𝑋, Σ, 𝜇)$ be a measure space. The Lebesgue space $𝐿 2 (𝑋, 𝜇)$ of complex-valued measurable and square-integrable functions¹ becomes a Hilbert space with the inner product

⟨ 𝑓 | 𝑔 ⟩ = \int 𝑋 𝑓 (𝑥) * 𝑔 (𝑥) 𝑑 𝜇 (𝑥)

Proof

#missing/proof

Dense bases

See Exponential Fourier series.

#state/develop | #lang/en | #SemBr

modulo functions that are equal almost everywhere. ↩