Some Algebraic and Topological Structures of Laplace Transformable Functions

Abstract

Abstract: In this work, the set of all functions that are Laplace transformable with regard to their structures both algebraic and topological, is taken into account. Certain topological properties of the set of Laplace transformable functions with the help of a metric are described. Also, we determine the proofs of the statements that the set of all Laplace transformable functions is a commutative semi-group with respect to the convolution operation as well as an Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Laplace transformable functions is defined and the proof that the Laplace transformable functions' space is complete with our metric is given. The separability theorem and that the Laplace transformable functions' space is disconnected are also discussed. Keywords: Abelian group, Commutative semi-group, Disconnected space, Laplace transform, Separability theorem.