It is a well-known result of Functional Analysis/Topological Vector space [Rudin(1991)] that every linear map from a finite dimensional normed space to any topological vector space E is continuous (=jointly continuous).
It is a well-known result of Functional Analysis/Topological Vector space [Rudin(1991)] that every linear map from a finite dimensional normed space to any topological vector space E is continuous (=jointly continuous).