# Banach and Hilbert spaces

thephystudent
As a european bachelor student in physics, i can follow a theoretical math course next year about banach and hilbert spaces. How useful are those subjects for physics?

Staff Emeritus
• In the beginning of QM, there were two popular formulations which seemed to work. Nobody knew which one was correct. It was eventually discovered that both theories are actually the same thing. This is because $\ell^2$ and $L^2$ are isomorphic Hilbert spaces.
• In QM, we interpret $|<a,b>|^2$ as some kind of probability. But why does this work? For example, why do those probabilities sum to 1? This is answered by the Parseval equality.