We seem to be quite familiar with Dirac Delta function, it’s like a spike.

In applying Dirac Delta to the initial understanding of defining a quantum state by a infinite combination of vectors/wave functions,

We need to locate the x that has to be exact the state here it is 2.71, hence it’s 0 everywhere else, but spike on 2.71.

The delta function could be less rigorous enough to the favor of mathematicians, but Brandon came up with below definition to justify it:

On this foundation,

SO leads to the dot product form again.