In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a good definition of it, and what are the possible difficulties of defining it.

We have a Hilbert space $\mathcal{H}$, and a family of "vectors", $ \left\lbrace|x \rangle \right\rbrace_{x\in \mathbb{R}} $ such that any vector $|\psi \rangle $ in $\mathcal{H}$ can be written

" $|\psi \rangle = \int_{\mathbb{R}} \psi(x)|x \rangle $ "

and

$\int_{\mathbb{R}} |x \rangle \langle x | dx = Id$

We don't say where those $|x \rangle $ live, and in general they are not in $\mathcal{H}$. Usual example from physics $\mathcal{H}=L^2(\mathbb{R})$, and where we think of $|x \rangle$ as the delta distribution.

It looks a little bit like the spectral theorem that allows to write self adjoint operators as integral with a projector valued measure on the spectrum. I also came across that word, "direct integral" that may have a link.

we usually also take "orthonormal basis", i.e $\langle x |y \rangle = \delta(x-y)$ (what is the meaning of seing it as a distribution in 2 variables)

Of course the decomposition of a vector has to be unique. In the example $\mathcal{H}=L^2(\mathbb{R})$ the coefficients is itself a function $\psi\in L^2(\mathbb{R})$, then unique in the sense "equal almost everywhere"

Among all the questions such presentation raises:

- is the hilbert space structure playing any role? example in defining $\langle x |$ as the "dual basis". Because since those vectors are not even in the hilbert space, their scalar product is not defined.
- is there any difficulty to define an abstract vector space generated by the family $ \left\lbrace|x \rangle \right\rbrace_{x\in \mathbb{R}} $. (Then of course there is still the problem of why a vector from our starting hibert space is equal to some construction in another abstract space.)
- If this whole business is actually well defined, we can see fourier transform as a particular case of writing a vector in the following generalized basis (also from physics)

$|p\rangle:= \left\lbrace x\rightarrow \frac{1}{\sqrt{2\pi}} e^{ipx}\right\rbrace$

and the theorems of Fourier transform being an isometry and being invertible would just say that $|p\rangle $ is an "orthonormale" basis.