Result: Spaces of analytic functions represented by Dirichlet series of two complex variables

Summary: We consider the space \(X\) of all analytic functions \[ f(s_1,s_2)= \sum^\infty_{m,n=1} a_{mn}\exp(s_1 \lambda_m+s_2 \mu_n) \] of two complex variables \(s_1\) and \(s_2\), equipped with the natural locally convex topology and using the growth parameter, the order of \(f\) is defined as recently by the authors. Under this topology \(X\) becomes a Fréchet space. Apart from finding the characterization of continuous linear functionals and linear transformations on \(X\), we obtain necessary and sufficient conditions for a double sequence in \(X\) to be a proper basis.