What works in separable Hilbert spaces that does not work in non-separable ones? 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.
 A: 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:

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into algorithms for use in numerical analysis […]

tl;dr:
The Galerkin method does not work in non-separable Hilbert spaces. 
