We know that dim(C[a,b]) is infinte. Indeed it cannot be finite since it contains the set of all polynomials. Is the dimension of a Hamel basis for it countable or uncountable? I guess if we put a norm on it to make a Banach space, we could use Baire's to imply uncountable. I am however interested in a proof that does not rely on equipping the vector space with a norm. If it's too long winded but available somewhere else(books, articles) kindly point it out to me and I'll read it.