Where precisely is topology required to prove the existence of non-zero eigenstates in the proof of the spectral theorem? My professor today remarked that the proof of the spectral theorem (even for the discrete spectrum case) uses not just algebra but also topology to prove the existence of eigenstates. However, I'm not being able to pinpoint which step of the proof of the spectral theorem (for infinite-dimensional operators) invokes topological arguments. Could someone please explain?

At the very least you need to establish that the spectrum is nonempty. This usually requires the Fundamental Theorem of Algebra in the finite-dimensional case, or some form of Liouville's Theorem in the general case. Examples:


*

*Theorem VII.3.6 in Conway's A Course in Functional Analysis.

*Theorem 1.2.5 in Murphy's C$^*$-Algebras and Operator Theory

*Theorem 3.2.3 in Kadison-Ringrose's Fundamentals of Theory of Operator Algebras

*Theorem 4.1.13 in Pedersen's Analysis Now

*Lemma VII.3.4.4 in Dumford-Schwartz Linear Operators
