Families of L_p-spaces with inductive and projective topologies

Henry Werner Davis, F. J. Murray and J. K. Weber

Vol. 34 (1970), No. 3, 619–638

DOI: 10.2140/pjm.1970.34.619

Abstract

Let (X,𝒜,μ) be a measure space, and S ⊂ [1,∞). This paper investigates basic properties of L^P(S) = ⋂ _t∈sL_t(μ) and L^I(S) = span of ⋃ _t∈sL_t(μ), when they are endowed with appropriate projective and inductive topologies.

If X is μ-finite or μ is a counting measure, then L^P(S), L^I(S) are projective and inductive limits in the usual sense. In this case the extensive abstract theory of inductive and projective limits applies. In the general case, however, this theory does not appear applicable. Using special properties of L_p-spaces a basic duality is established between L^P(S) and L^I(S′), for the general case, where S^γ is the set of conjugates to elements of S.

Next such properties as metrizability, normability and completeness for L^P(S),L^I(S) are considered. The question of when L^P(S) = L^P(T) is also considered, and it is shown that there is a certain maximal set T for which this is true. Similarly for L^I(S).

Mathematical Subject Classification

Primary: 46.35

Secondary: 28.00

Milestones

Received: 4 December 1969

Published: 1 September 1970