  Hardy spaces in the complex plane and in higher dimensions have natural
finite-dimensional subspaces formed by polynomials or by linear maps. We use
the restriction of Hardy norms to such subspaces to describe the set of
possible derivatives of harmonic self-maps of a ball, providing a version of
the Schwarz lemma for harmonic maps. These restricted Hardy norms display
unexpected near-isometric duality between the exponents 1 and 4, which we use
to give an explicit form of harmonic Schwarz lemma.
