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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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    $\begingroup$ An example of a non-separable Hilbert space is $L^2$, the space of square integrable functions. This space is widely applicable in quantum mechanics and probability theory. $\endgroup$
    – user275377
    Sep 27, 2017 at 22:20
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    $\begingroup$ @Alex $L^2$ is separable since $L^2([0,1])$ is separable, see also the Hermite functions. Now it is not a RKHS $\endgroup$
    – reuns
    Sep 27, 2017 at 22:21
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    $\begingroup$ @reuns separability of $L^2(\Omega, \mathcal{F}, \mu)$ depends on the choice of measure space $(\Omega, \mathcal{F}, \mu)$. $L^2([0,1])$ (with the lebesgue measure and borel $\sigma$-algebra) is separable but that has nothing to do with separability of other $L^2$-spaces. $\endgroup$
    – Rhys Steele
    Sep 27, 2017 at 22:31
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    $\begingroup$ @reuns You're free to call whatever you want an $L^2$ space. Unfortunately here you disagree with common usage since $([0,1], P([0,1]), \mu)$ is a measure space against which we have a notion of the Lebesgue integral. $\endgroup$
    – Rhys Steele
    Sep 27, 2017 at 22:58
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    $\begingroup$ @nobody Locally finite means Integrating locally constant functions makes sense. Otherwise it becomes very different so at the very least you should mention that it is. $\endgroup$
    – reuns
    Sep 27, 2017 at 23:02

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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.

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    $\begingroup$ Thanks a lot. I can connect to this example much better and now I can see how to frame it for my own problem. Also, your answer should be correct too, but I think I can only pick one. $\endgroup$
    – user1609656
    Sep 28, 2017 at 18:51
  • $\begingroup$ Instead of $l^2(\mathbb R)$ I would say $l^2(S)$ where $S$ is an arbitrary uncountable set. Also, the functions $f_y$ form a base which means the dimension of a Hilbert space can be any cardinal number. $\endgroup$
    – Heimdall
    Jan 24, 2019 at 9:10
  • $\begingroup$ What I wonder, though, is: has there been any research on non-separable / uncountably-dimensional Hilbert spaces? In particular, we know that for any countable cardinal number $k$ (so finite ones and $\aleph_0$) there is (up to isomorphism) exactly one $k$-dimensional Hilbert space. Is it so for uncountable cardinals, too? The $l^2(S)$ (where $S$ is some set of cardinality $k$) contruction only tells there is at least one $k$-dimensional Hilbert space. $\endgroup$
    – Heimdall
    Jan 24, 2019 at 9:22
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    $\begingroup$ @daw: I'm not sure I get your argument for why $l^2(\mathbb R)$ is not separable.True that countable union of countable sets is countable but $\mathbb R$ is uncountable but separable ( by being 2nd countable). $\endgroup$
    – MSIS
    Sep 13, 2020 at 19:46
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    $\begingroup$ Separability of $\mathbb R$ does not matter here/does not help. The uncountability of $\mathbb R$ is the property that gives non-separability of $l^2(\mathbb R)$. $\endgroup$
    – daw
    Sep 14, 2020 at 8:15
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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.

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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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