It is fairly common-place amongst physicists to consider physical observables that may take a continuum of values $x$ and follow Dirac in representing a physical state as an integral over a set of eigenstates $\left|x\right>$ belonging to these eigenvalues, such as
\begin{equation} \left|\Psi\right> = \int \left|x\right>\left<x|\Psi\right>~dx. \end{equation}
This implies that the integral
\begin{equation} \int \left|x\right>\left<x\right|~dx = \hat{I} \end{equation}
plays the role of the identity operator on the space of the $\left|x\right>$.
However, since the eigenvalues $x$ take a continuous range of real numbers and the $\left|x\right>$ are mapped to these in a one-to-one relation, the $\left|x\right>$ must form an uncountably infinite orthonormal basis (since the real numbers are uncountably infinite). This goes beyond the limits of a Hilbert space, which, according to von Neumann [1], must be of either finite or countably infinite dimension.
Dirac, himself, comments in [2] that:
The space of bra or ket vectors when the vectors are restricted to be of finite length and to have finite scalar products is called by mathematicians a Hilbert space. The bra and ket vectors that we now use form a more general space than a Hilbert space.
Dirac then goes on to define his famous delta for dealing with eigenstates with continuous eigenvalues - a method that von Neumann severely criticises, arguing that this approach lacks mathematical rigour [1].
My question is then: have pure mathematicians consistently defined a type or types of complex vector space that can have uncountably infinite dimensions that are appropriate for Dirac's approach? If so, (for the purposes of further reading) what do they call such vector spaces? Moreover, if such vector spaces can be consistently defined, is it possible to make Dirac's approach mathematically rigorous or do we need to consider expressions such as the above involving integrals of eigenstates as spurious?
[1] Mathematical Foundations of Quantum Mechanics (New Edition), John von Neumann, Princeton University Press (2018)
[2] The Principles of Quantum Mechanics, Fourth Edition, P.A.M. Dirac, Clarendon Press, Oxford (1999)
