In this note we consider the problem of localization and approximation of eigenvalues of operators on infinite dimensional Banach and Hilbert spaces. This problem has been studied for operators of finite rank but it is seldom investigated in the infinite dimensional case. The eigenvalues of an operator (between infinite dimensional vector spaces) can be positioned in different parts of the spectrum of the operator, even it is not necessary to be isolated points in the spectrum. Also, an isolated point in the spectrum is not necessary an eigenvalue. One method that we can apply is using Weyl's theorem for an operator, which asserts that every point outside the Weyl spectrum is an isolated eigenvalue.