  In this paper we develop a version of spectral theory for bounded linear
operators on topological vector spaces. We show that the Gelfand formula for
spectral radius and Neumann series can still be naturally interpreted for
operators on topological vector spaces. Of course, the resulting theory has
many similarities to the conventional spectral theory of bounded operators on
Banach spaces, though there are several important differences. The main
difference is that an operator on a topological vector space has several
spectra and several spectral radii, which fit a well-organized pattern.
