Hilbert-substructure of Real Measurable Spaces on Reductive Groups, I; Basic Theory (original) (raw)

This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, F n (G), of real L p (G)−spaces (when G is a reductive group in the Harish-Chandra class and p=2n), the situation may be rectified, via an outlook which generalizes the fine structure of the Hilbert space, L 2 (G). This success opens the door for harmonic analysis of unitary representations, G→End(F n (G)), of G on the Hilbert-substructure F n (G), which has hitherto been considered impossible.