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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?

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  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/27665/2451 , physics.stackexchange.com/q/234/2451 and links therein. $\endgroup$ – Qmechanic Mar 30 '14 at 19:17
  • $\begingroup$ Hi @NeuroFuzzy, this question fits poorly on Phys.SE for various reasons e.g. it tends to be primarily opinion-based. I'm closing this as a duplicate, not because it is an exact duplicate, but to guide you in the right direction. $\endgroup$ – Qmechanic Mar 30 '14 at 20:10
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    $\begingroup$ Peruse the four volumes by Reed and Simon, Methods of modern mathematical physics. Then you can form your own opinion. $\endgroup$ – Urgje Mar 31 '14 at 9:43