Where I can find (in which book) proof of the following theorem:
"The spectrum of a function in the algebra is the set of values of the function"
or how can I proof it?
If C(S) is the Banach algebra of continuous functions on a compact set S, and , the inverse of will be the function g given by g(t) = (provided that such a function exists). If f(t) is never equal to then g will be defined on all of S, bounded (because S is compact) and continuous. Therefore has an inverse in C(S) and hence . But if there is a value of t for which then g is not defined at that value of t. In that case has no inverse in C(S) and hence .