Isometries of the spaces of bounded frame functions

We define a natural metric on the space of all bounded frame functions on a given Hilbert space and also on the space of all bounded frame functions with a given weight. We present the complete description of the surjective isometries of the so-obtained metric spaces. In fact, it turns out that those isometries are intrinsically induced by unitary or antiunitary operators on the underlying Hilbert space.