The classical Burnside's Theorem guarantees in a finite dimensional space the existence of invariant subspaces for a proper subalgebra of the matrix algebra. We give an extension of Burnside's Theorem for a general Banach space, which implies general results on invariant subspaces. We also show that any weakly closed algebra of bounded operators acting on a Banach space and different from the algebra of all bounded operators admits positive vector-functionals continuous in the essential operator norm.