Is every Hilbert space separable?
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?
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$\begingroup$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.
$\endgroup$ 5 $\begingroup$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.
$\endgroup$ 0 $\begingroup$Yes, some researchers have worked on Fredholm theory using non separable Hilbert spaces. For instance you may go through Approximation by semi-Fredholm and semi-alpha-Fredholm Operators in Hilbert spaces of arbitrary dimension by Laura Burlando in Acta Sci.Math(Szeged) 65(1999).
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