Given a normed vector space X, the norm gives rise to a metric that satisfies
1. d(x + z, y + z) = d(x, y) and 2. d(ax, ay) = |a|d(x, y) for all x, y, z in X.
I'm looking for a vector space X with a metric that does not satisfy these. I've found that |x^3 - y^3| on R does not satisfy either. I can't think of any that satisfy 1 but not 2, and 2 but not 1.