I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions".

Given a finite dimensional vector space with a basis $\{x_i\}$, I understand that the basis is orthonormal if $\langle x_i|x_j\rangle = \delta_{ij}$ (where $\delta_{ij}$ is the Kronecker delta).

Extending this to continuous infinite dimensions, an example "basis" vector (whatever this really means in this case) may look like $|x\rangle$ where $x$ can be any real number in some interval. Apparently the basis vectors $|x\rangle$ and $|y\rangle$ are orthonormal if $\langle x|y\rangle = \delta(x-y)$, where $\delta(x-y)$ is the Dirac delta.

I would be convinced of this generalization if I could show the following:

  • Define an inner product in the finite dimensional case with a basis satisfying the first orthonormality condition.
  • Let the number of dimensions somehow go from finite to a continuous infinity.
  • Show that in this limit basis still satisfies the second orthonormality condition.

Can the above be done, or is there a fundamental misconception I have here? Do you have any recommended reading? (preferably not an Analysis textbook)