Fr\'echet spaces of general Dirichlet series

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