I'm an undergraduate physics without much quantum mechanics at all under my belt. I'm studying functional analysis, and I want to know whether or not this will be useful for me in theoretical physics in the future. Some perceived benefits of studying functional analysis are:
- It gives familiarity with spaces (Hilbert spaces especially) and linear operators in their most general form, which (so I hear) pop up all the time in quantum mechanics.
- I have noticed that the treatment of linear forms, bilinear forms, and duals, in pure functional analysis, complement what Penrose talks about in Road to Reality, where the dual of a vector is treated as a function which takes in a vector and spits out a real number. This is "weird" because I used to think of the dual to a vector as essentially another vector, but Functional analysis seems to spell this out and give all the needed isomorphisms to make sense of it all, in whatever form desired.
- It defines distributions in their most general form, which seems especially useful to make sense of, say, $\delta'$ and $\delta''$, where $\delta$ is the Dirac delta, should they arise in physics while solving a differential equation.
But of course, those three things could have been learned, possibly in a shorter amount of time, by reading from a physics book and taking a less "definition-theorem-proof" approach to the whole subject. On the other hand, the rigor, I think, might help me identify which assumptions are physical, which are definitions, and which are mathematical theorems.
But I'd like to ask: In what other ways is rigorous functional analysis useful for theoretical physics? And are there any other ways that it isn't useful?