How to proof a basis ${\psi_a}$ is complete? Why $$\int\text{d}a\psi^*_a(y)\psi_a(x)=\delta(y-x)$$
shows the basis is complete?
Even, how is $\delta$ defined? I mean, the most consistent definition.
I really dislike the definition by 0 and $\infty$.
 A: A way to interpret the delta function in this context is through an integral. Start with the example of the Fourier transform composed with its inverse.
$$
              f = \frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}f(t)e^{-ist}dt\right)e^{isx}ds.
$$
This may be correctly written as
$$
       f(x)=\lim_{R\rightarrow\infty}\int_{-\infty}^{\infty}f(t)\left(\frac{1}{2\pi}\int_{-R}^{R}e^{-ist}e^{isx}ds\right)dt.
$$
But what you can't do using standard methods is to interchange the limit with the other intergral in order to write this as
$$
       f(x) = \int_{-\infty}^{\infty}f(t)\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-ist}e^{isx}ds\right)dt
$$
That's what Dirac is doing. And, honestly, there's not much harm in doing this so long as you understand what is really meant.
In your case, you could understand the expression to mean
$$
       f(x)=\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty} f(y)\psi_{a}^*(y)dy\right)\psi_{a}(x) da.
$$
That can have perfectly sound meaning, even though it is not as general as it needs to be in order to handle even the basic cases of Quantum in 1d, where a couple of extensions are already needed:


*

*
*

*There can be a mix of discrete and continuous spectrum, requiring integrals and sums.


*

*For periodic 1-d problems the dimensions of the eigenspaces can be 1 and or 2, requiring more than a single integral.



When you then try to extend the theory to 3d and beyond, Dirac's formulation is really not sufficient for a proper theoretical framework, and patches to try to keep the old formalism hobbling along become increasingly obscure, if not nightmarish.
Dirac's formalism works well for ODEs on finite or infinite intervals $(a,b)$ where any required endpoint conditions are separated (not periodic.) In such cases of ODEs, Dirac knew from Weyl and others that you could write
$$
                f = \int_{-\infty}^{\infty}\left(\int_{a}^{b}f(t)\varphi_{\lambda}(t)w(t)dt\right)\varphi_{\lambda}(x)d\rho(\lambda). \tag{$\dagger$}
$$
The $\varphi_{\lambda}(t)$ are classical solutions of the eigenvalue equation for the ODE. You need a weight function $w$ in the inner integral, and you need a spectral density measure $\rho(\lambda)$. The outer integral may be viewed as a Riemann-Stieltjes or probability distribution type of integral. The convergence of the integrals needs to be discussed, but it certainly holds in $L^2$. Here Dirac's notation is workable and compelling, and the basic cases of Quantum mostly all fit into this framework. In this framework, Dirac would write
$$
          \int_{-\infty}^{\infty}\varphi_{\lambda}(x)\varphi_{\lambda}(y)d\rho(\lambda) = \delta(x-y)
$$
in order to summarize the previous equation $(\dagger)$
