The algebra A(E)

Given a locally convex algebra E, we shall construct a vector space A(E) on which multiplication is defined on it. We will show the multiplication so defined on A(E) is associative and that there is a natural embedding of the sets E and its dual space E* in to the vector space A(E). This embedding allows to carryover the operations (linear and bilinear) defined on E and E* to the algebra A(E). Finally we discussed and derive some results about A(E) when E = S(R), the space of functions of rapid decay.