# What is the relationship between completeness of wave functions and completeness of Hilbert space?

In the lecture, my prof said that completeness means that any wave function can be constructed using an infinite number of "other" basis wave functions. This is very intuitive since this is nothing but a Fourier series.

But then each wave function is a vector on Hilbert space, and completeness on Hilbert space means that there exists a cauchy sequence such that $\lim_{n\to\infty} f_n - f_m = 0$

Are there connections between completeness of Hilbert space and completeness of wave functions?

• It is also worth stressing that, for vector spaces equipped with a (positive) scalar product, the two notions are equivalent provided the distance is the one defined by the scalar product $d(x,y) := \sqrt{(x-y,x-y)}$ and the considered bases are the orthonormal ones. Feb 24, 2015 at 17:26