>>12261783in an analysis class, some universities call it "Linear Analysis" (it's technically Functional Analysis, but that's more of a grad course).
Hilbert spaces are a kind of infinite dimensional vector space. in finite dimensional case, R^n or C^n are prototypical vector spaces, and everything is isomorphic to one of them (depends on if the vector space is real or complex). in infinite dimensions, there are lots of different weird vector spaces that are incompatible (depends on the kinds of features you want them to have). Hilbert spaces are ones with super duper nice properties, they act as good infinite dimensional version of finite dimensional vector spaces. they also have Hilbert-space bases of certain cardinalities, and any two with the same cardinality are Hilbert-space isomorphic.
