Hilbert space

Definition

A Hilbert space is a vector space, $(H,F)$ (where $F$ is either $\mathbb{R}$ or $\mathbb{C}$) with an Inner product $$\langle\cdot,\cdot\rangle$$ such that $H$ is complete with respect to the associated norm $$\|x\|=\sqrt{\langle x,x\rangle}$$

That is to say a Hilbert space is a Banach space where the norm is given by an inner product