# Hilbert Space

## Definition

A Hilbert Space, \(\mathcal{H}\) is a real or complex inner product space and a complete metric space with respect to the distance function induced by the inner product.

Another formulation is that the Hilbert space is an inner product space that is separable and complete.

Yet another formulation is that the Hilbert space is a Cauchy Complete Separable Inner Product Space.

### Consequence

It is Cauchy Complete.

### Theorem

All separable Hilbert spaces are isomorphic to \(\ell_2\).