Let V be an inner product space whose inner product g is not positive-definite but simply nondegenerate (Minkowski spacetime is an example). Using Zorn's Lemma I've proven fairly easily that V has a maximal subspace on which g is negative-definite. Now I'm investigating: Conjecture: V has a subspace, of maximal dimension, on which g is negative-definite. If V is finite-dimensional, this is obviously true. Do you guys think it is true if V is of infinite dimension?