Semi-groups of operators and the Weierstrass theorem

In this note we give a short proof of the generalization of M. H. Stone's representation of groups of unitary operators in Hubert space and show how it yields the theorem of Weierstrass on uniform approximation by polynomials. This classical result in turn yields fairly easily the more recent algebraic-topological formulation of the Weierstrass theorem as given by M. H. Stone for real algebras and by I. Gelfand and G. Silov for complex algebras. For 0 ^ 5 < oo let T8 be a linear operation in the real or complex Banach space X satisfying the conditions