Abstract
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.
Highlights
Let G be a reductive group in the Harish-Chandra class and denote Lp(G, ) as the Lebesque spaces of −valued functions on G, 0
As known for L2(G, ), could be said about other members of the Lebesque spaces on G. This is largely due to the absence of a manageable ‘Hilbert space theory’ (which made the discussion of unitary representations of G on Lp>2(G, ) a forbidden concept) and has led to the employment of indirect techniques to extract important results out of them
We introduce the notion of orthogonality in Fn(G)
Summary
Let G be a reductive group in the Harish-Chandra class and denote Lp(G, ) as the Lebesque spaces of −valued functions on G, 0
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