Fra\"iss\'e and Ramsey properties of Fr\'echet spaces

We develop the theory of Fra\"iss\'e limits for classes of finite-dimensional multi-seminormed spaces, which are defined to be vector spaces equipped with a finite sequence of seminorms. We define a notion of a Fra\"iss\'e Fr\'echet space and we use the Fra\"iss\'e correspondence in this setting to obtain many examples of such spaces. This allows us to give a Fra\"iss\'e-theoretic construction of $(\mathbb{G}^\omega, (\|\cdot\|_n)_{n