The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.
This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.
The theorem can be extended to give a characterization of weakly compact convex sets.
Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.
Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.