hilbert space

A Hilbert space is a vector space $H$ equipped with an inner product $⟨\cdot ,\cdot ⟩$, such that every (Cauchy) sequence is convergent (i.e., the limit exists within $H$), where convergence is measured by the distance given by $||f||=\sqrt{⟨f,f⟩}$.

For instance, the space

$$L_2(\mathcal{X})=\{m:\mathcal{X}\to \mathbb{R}|\int m^2(x)dx<\infty \},$$

with $⟨f,g⟩=\int f(x)g(x)dx$ is a Hilbert space.

Hilbert spaces are strict subsets of Holder spaces. Any Hilbert space is also a (2,1)-smooth Banach space.