Application of the Stone-Weierstrass Theorem to the Creation of Complete Sequences in Spaces of Integrable Functions

In this paper, using the Stone-Weierstrass theorem, a theorem is proved which affords possibilities for the easy formation of complete sequences in spaces of integrable functions. Moreover, sufficient conditions of the completeness of sequences in the energetic space of a positive definite operator are determined. A few examples are given, which indicate manners in which the theorem proved can be used to create complete sequences both in the space of quadratically integrable functions and in the energetic space of the operator -laplacian with homogeneous Dirichlet boundary condition.