Let R^oo denote the real vector space that is the countable direct product of copies of the reals.
I.e., all countable-tuples of reals with componentwise addition.
Puzzle: What is the dimension of the real vector space R^oo ???
I thought about the same question, in a slightly more general setting a few years ago. Specifically, let k be a field, and let Map(N, k) be the k-vector space of all functions from N (the set of natural numbers) to k . The question I considered, was "what is the dimension of Map(N, k) over k ?" I found it very counterintuitive that the answer depends upon the field k ! I did not solve it in all cases, but I can do Dan's case, in which case the dimension is c = 2^Aleph_0 . In general, the dimension is at least max(|k|, c) . I do not know if this is the right answer if k has cardinality strictly between Aleph_0 and c (in which case CH is false). For x in k , consider the function n |--> x^n . These functions are linearly independent over k , so the dimension is at least |k| . On the other hand, the vector space V = Map(N, k) has cardinality at least c . If k is an infinite field, and B is a basis of V , then |V| = |B| |k| = max(|B|, |k|) . This shows that if |k| < c , then |B| >= c . If k is finite, then it's easy to see that |V| = c , and B must be infinite, in which case |V| = |B| , so the dimension is c . (When k = R , the vector space has cardinality c , so a basis can't have cardinality larger than c .) Michael Reid