Vol. 34, No. 3, 1970

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Families of Lp-spaces with inductive and projective topologies

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

Vol. 34 (1970), No. 3, 619–638

Let (X,𝒜) be a measure space, and S [1,). This paper investigates basic properties of LP(S) = tsLt(μ) and LI(S) = span of tsLt(μ), when they are endowed with appropriate projective and inductive topologies.

If X is μ-finite or μ is a counting measure, then LP(S), LI(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 Lp-spaces a basic duality is established between LP(S) and LI(S), for the general case, where Sγ is the set of conjugates to elements of S.

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

Mathematical Subject Classification
Primary: 46.35
Secondary: 28.00
Received: 4 December 1969
Published: 1 September 1970
Henry Werner Davis
F. J. Murray
J. K. Weber