Vol. 53, No. 1, 1974

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Theorems of Korovkin type for Lp-spaces

S. J. Bernau

Vol. 53 (1974), No. 1, 11–19

Suppose (X,Σ) is a measure space, 1 < p < ,p2, and that (Tn) is a net of linear contractions on (real or complex) Lp(X,Σ). Let M = {x Lp : Tnx x} (M is the convergence set for (Tn)). It is obvious that M is a closed subspace of Lp; indeed this would be true for an arbitrary normed space. In this paper we shall show that M is the range of a contractive projection on Lp and hence is itself isometrically isomorphic to an Lp-space. If S Lp(X,Σ) we can define tke shadow, 𝒮(S) of S to be the set of all x in Lp such that Tnx x for every net of linear contraclions (Tn) such that Tny y for all y S. We shall also give a complete description of 𝒮(S) (for p1,2,).

Mathematical Subject Classification 2000
Primary: 41A65
Received: 21 June 1973
Published: 1 July 1974
S. J. Bernau