ABSTRACT
In this chapter we depart from the general theme of the volume, which has featured spaces of vector-valued functions. Here we are concerned with the size of the group of isometries on a given space.
If x, y are elements on the surface of the unit ball of a Hilbert space X, then it is well known that there is a surjective isometry T of the space such that Tx = y. The norm of a Banach space which has this property is said to be transitive. An equivalent way to describe transitivity of the norm is to say that for each x ∈ X with ‖x‖ = 1, the orbit G(x) = {Tx : T ∈ G} is equal to the surface S(X) of the unit ball of X, where, as usual, G = G(X) denotes the group of surjective isometries on (X, ‖ · ‖). A question that has remained open since the time of Banach asks whether a separable Banach space with a transitive norm must be a Hilbert space.
