Why is the "complete metric space" property of Hilbert spaces needed in quantum mechanics?

I have been learning more about Hilbert spaces in an effort to better understand quantum mechanics. Most of the properties of Hilbert spaces seem useful (e.g. vector space, inner product, complex numbers, etc), but I don't understand why we need the inner product space to form a complete metric space.

My Questions:

1. Why do we need the inner product space to form a metric space?


2. Why does it have to be a complete metric space?