On the Geometry of Dual Pairs

The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non-zero vector has a positive multiple which is the sum of two v-dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc-connectedness of the numerical range determined by v.