This paper reconsiders the age-long problem of normed linear spaces which do not admit inner product and shows that, for some subspaces, Fn(G), of real Lp(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, L2(G). This success opens the door for harmonic analysis of unitary representations, G→End(Fn(G)), of G on the Hilbert-substructure Fn(G), which has hitherto been considered impossible.
"Hilbert-substructure of Real Measurable Spaces on Reductive Groups, I; Basic Theory." J. Gen. Lie Theory Appl. 10 (1) 1 - 4, 2016. https://doi.org/10.4172/1736-4337.1000242