Fréchet spaces of general Dirichlet series

Inspired by a recent article on Fréchet spaces of ordinary Dirichlet series $$\sum a_n n^{-s}$$ due to J. Bonet, we study topological and geometrical properties certain scales general e^{-\lambda _n s}$$ focus the space $$\lambda $$ -Dirichlet which have limit functions bounded all half planes strictly smaller than right plane $$[{{\,\mathrm{Re}\,}}>0]$$ . We develop an abstract setting pre-Fréchet generated admissible normed abscissas convergence they generate, allows also define for $$a_n _n/k}$$ each k equals Fourier coefficients function appropriate group.