On the Geometry of Dual Pairs
Bryan E. Cain, B. David Saunders, Hans Schneider · Studies in Applied Mathematics · 1977
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.