For finite-dimensional space X, we know any operator T on X can be seen as a matrix. From there we can give the existence of eigenvalues through the fundamental theorem of algebra. The set of all eigenvalues will be the spectrum of T. Clearly it is a compact set and its complement in the complex plane is the resolvent set of T. If I am wrong anywhere then please correct me. Thank you in advance.
2026-04-11 16:51:53.1775926313
How to prove the existence of eigenvalues of T if X is an infinite-dimensional space, where T is an operator on X?
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