# Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]

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)