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