Hamel basis for V. (A Hamel basis is a maximal linearly independent set. The existence of such a set will usually require the Axiom of Choice.) Then every element of V has a unique expression as a finite linear combination of basis elements.
So, for each x in V there is a unique expression , where only finitely many of the coefficients are nonzero. Define That defines a norm on V.
That construction works in particular when V is the space of all continuous functions on .