Fraïssé and Ramsey properties of Fréchet spaces

We develop the theory of Fraïssé limits for classes finite-dimensional multi-seminormed spaces, which are defined to be vector spaces equipped with a finite sequence seminorms. define notion Fréchet space and we use correspondence in this setting obtain many examples such spaces. This allows us give Fraïssé-theoretic construction (Gω,(‖⋅‖n)n<ω), separable almost universal disposition class all an infinite then identify prove approximate Ramsey property various using known properties normed A version Kechris-Pestov-Todorčević approximately ultrahomogeneous is also established used new extremely amenable groups. In particular, show that group surjective linear seminorm-preserving isometries (Gω,(‖⋅‖n)n<ω) amenable.