Does a continously infinite vector space make sense? Does the concept of a continuously infinite vector space make sense? That is, our dimension number $k$ has a totally continuous range as opposed to $n\in (1,2,3,...)$.
The context behind this are Fourier series and transforms. In the series case, we might have a basis $\psi_n(x)$ ( orthonormal even ), and (almost ) any function has a series representation with respect to this basis. However, when the integer index $n$ is replaced by the continuous index $k$, the representation is achieved with a Fourier transform instead. Our basis $\psi_k(x)$ is still infinite, but it is not countable. Also, in the series case the linear weights are $c_n$, but in the continuous case this thing is replaced by $\phi(k)$. The two situations are not entirely analogous.
 A: It does make sense, the dimension of a Hilbert space can be any cardinality, including the continuum, but that is not the sense that is at play in the Fourier transform examples. It is not the vector space that is continuously infinite there, but the "basis". The space itself can still be separable, i.e. admit a genuine basis of countable cardinality, and $L^2(\mathbb{R})$ does admit such a countable basis (of Hermite functions, for example). But it also admits continuous "bases", like  $e^{i \omega x}$ for $\omega\in\mathbb{R}$, and Fourier transforms are expansions in it.
The reason I put "basis" into scare quotes is that it is not actually a basis because $e^{i \omega x}\notin L^2(\mathbb{R})$, and so they do not belong to the space they are supposed to form a basis of. Nonetheless, it is possible to make sense of them as a basis by extending the original Hilbert space, or "rigging" it, as it is called, to a larger Hilbert space of distributions that includes $e^{i \omega x}$, and then expanding elements of the original space in this extension, see Rigged Hilbert space. The general theory is described, for example, in Functional Analysis by Berezansky-Sheftel-Us, v.II, ch.14 with applications to spectral analysis of differential operators.
This construction was invented by Gelfand (hence the alternative name for the rigging, Gelfand triple) to cover spectral decompositions in the case of continuous spectrum, a far reaching generalization of Fourier analysis. It is particularly popular in applications to quantum mechanics, where $e^{i \omega x}$ are the "eigenfunctions" of the momentum operator. Decompositions with respect to delta-functions (the "eigenfunctions" of the position operator) can also be handled the same way. This can be generalized even further, and one can take continuous integrals of families of Hilbert spaces to decompose infinite dimensional linear operators rather than elements, see Direct integrals of Hilbert spaces.
