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 of certain scales of Fréchet spaces of general Dirichlet spaces $\sum a_n e^{-λ_n s}$. More precisely, fixing a frequency $λ= (λ_n)$, we focus on the Fréchet space of $λ$-Dirichlet series which have limit functions bounded on all half planes strictly smaller than the right half plane $[\mathrm{Re} >0]$. We develop an abstract setting of pre-Fréchet spaces of $λ$-Dirichlet series generated by certain admissible normed spaces of $λ$-Dirichlet series and the abscissas of convergence they generate, which allows also to define Fréchet spaces of $λ$-Dirichlet series for which $a_n e^{-λ_n/k}$ for each $k$ equals the Fourier coefficients of a function on an appropriate $λ$-Dirichlet group.