What is a basis for the vector space of continuous functions?

Question

A natural vector space is the set of continuous functions on $\\mathbb{R}$. Is there a nice basis for this vector space? Or is this one of those situations where we're guaranteed a basis by invoking the Axiom of Choice, but are left rather unsatisfied?

Answer 1

By Zorn's lemma every vector space has a basis. The set of continuous real-valued functions is a vector space of infinite dimension. So there is not an explicit base in terms of a finite set of basic vectors. The polynomials are dense in this space, So a Schaude -basis for the space of continuous function might be the Taylor Series expansion of a continuous function or the Fourier Series expansion.