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This article gives a new proof of the fundamental lemma of the "weaklyadmissible implies admissible" theorem of Colmez-Fontaine that describes thesemi-stable p-adic representations. To this end, we introduce the category ofspectral Banach spaces, which are p-adic Banach spaces with a C_p-algebra ofanalytic functions, and the subcategory of effective Banach-Colmez spaces. Thefundamental lemma states the surjectivity of certain analytic maps anddescribes their kernel. It is proven by an explicit count of solutions of theequations defining these maps. It is equivalent to the existence of functionsof dimension and height of effective Banach-Colmez spaces.

Analytic p-adic Banach spaces and the fundamental lemma of Colmez and Fontaine