Is every Hilbert space separable?

Question

A Hilbert space is a complete inner product space; that is any Cauchy sequence is convergent using the metric induced by the inner product.

From Wikipedia: A Hilbert space is separable if and only if it has a countable orthonormal basis.

What are the examples of non-separable Hilbert spaces? From an applied point of view, are all interesting (finite or infinite) Hilbert spaces separable?

Answer 1

The set of almost periodic functions with the inner product $$\langle f, g \rangle = \lim_{N \to \infty} \frac{1}{2N} \int_{-N}^N f(x) \overline{g(x)}dx$$ has an uncountable orthonormal family $\{e^{i \omega x}\}_{\omega \in \mathbb{R}}$. Its completion is a non-separable Hilbert space.

Answer 2

The space $l^2(\mathbb R)$ is another example of a non-separable Hilbert space: It consists of all functions $f:\mathbb R\to\mathbb R$ such that $f(x)\ne0$ only for countable many $x$, and $$ \sum_{x\in \mathbb R}f(x)^2 <\infty. $$ It is easy to see that this is a Hilbert space, the crucial argument is that the countable union of countable sets is countable.

The functions $f_y$ defined by $$ f_y(x) = \begin{cases} 1 &\text{ if } x=y\\ 0 & \text{ else}\end{cases} $$ are an uncountable set of elements with distance $\sqrt2$, hence $l^2(\mathbb R)$ is not separable.