  We give a new proof of the "weakly admissible implies admissible" theorem of
Colmez and Fontaine describing the semi-stable p-adic representations. We study
Banach-Colmez spaces, i.e. p-adic Banach spaces with the extra data of a
C_p-algebra of analytic functions. The "weakly admissible" theorem is then a
result of the existence of dimension and height functions on these objects.
Furthermore, we show that the subcategory of Banach-Colmez spaces corresponding
to crystalline representations is naturally filtered by the positive rationals.
