When dealing with Hilbert spaces, we often restrict our attention to separable Hilbert spaces over non-separable ones. The need for separable Hilbert spaces is often explained with mathematical convenience, e.g. here:
"I have heard that usual mathematical manipulations that we take for granted will no longer hold [for non-separable Hilbert spaces]."
However, I have yet to find a list of mathematical manipulations that do not hold in non-separable Hilbert spaces.
What mathematical manipulations do not work for non-separable Hilbert spaces, and why?
To convey the spirit of the question, here are some mathematical manipulations that could potentially fail:
- Gram-Schmidt procedure
- Eigendecomposition
- Inverting (linear) maps
- (Construction of) dual spaces
I would also greatly appreciate a resource that discusses these problems in more detail.
There is no countable complete orthonormal system, so you can't choose a system $\{u_k\}$ such that you can write every element $u$ as $$ u =\sum_{k=1}^\infty \langle u,u_k\rangle u_k. $$ As Eric Wofsey commented, there is a similar formulation with uncountably many orthonormal $u_k$, so that technically, separability is not really special in this regard.
However, I'd like to add, that a countable orthonormal basis introduces a coordinate system and the Hilbert space really feels like "$\mathbb{R}^n$ with (countable) infinite $n$". While it may be true that, technically, a non-separable Hilbert space is, in the same sense, an "$\mathbb{R}^n$ with uncountable infinite $n$", one should use more caution when working in them, while working in the separable case really is, in practice very much like working in $\mathbb{R}^n$.
What is a notable difference between separable and non-separable Hilbert space is that discretization (i.e. approximating infinite dimensional problems with finite dimensional ones) is different: In a separable Hilbert space you can choose some orthonormal bases and approximate any $u$ in your problem by $u = \sum_{k=1}^N a_k u_k$ and try to reformulate your problem in the variables $a_k$ and solve that. In a non-separable Hilbert space this is different and I am not aware of a general approximation procedure that works in practice (i.e. can be implemented straight away).
The Wikipedia page on separability lists the same issue in a slightly different way:
tl;dr:
The Galerkin method does not work in non-separable Hilbert spaces.