I am having trouble accepting two well known results of analysis as non contradictory. First, given a vector space equipped with norm ||.||, the unit ball is compact iff the space is finite dimensional. Second, the Arzela-Ascoli theorem asserts that given a compact set X, a set S contained in the space of all continuous functions f(X) equipped with the supremum norm is compact iff it is uniformly closed, uniformly bounded and equicontinuous. My problem is the following: let S be the set of all equicontinuous functions on X with ||f|| <= 1, where ||.|| is the supremum norm. Is the space of all equicontinuous functions not an infinite dimensional subspace? Yet S is compact. I realize there is something fundamental I'm not getting. Please help!