Hilbert Spaces

Definition: A Hilbert Space is an inner product space $H$ that is a Banach space with respect to the norm induced by the inner product.

So every Hilbert space is a Banach space.

The simplest example of a Hilbert space is $\mathbb{R}^n$ and $\mathbb{C}^n$. The inner products on these spaces is the Euclidean dot product given by:

(1)

\begin{align} \quad \langle x, y \rangle = \sum_{k=1}^{n} x_k\overline{y_k} \end{align}

It is easy to verify that the norm induced by the Euclidean dot product is the usual Euclidean norm on $\mathbb{R}^n$ and $\mathbb{C}^n$ which we know is complete and is hence a Banach space.