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The range of a representing measure

Published online by Cambridge University Press:  01 September 1998

PAUL W. LEWIS  and
JAMES P. OCHOA
PAUL W. LEWIS
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas U.S.A.
JAMES P. OCHOA
Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas U.S.A. Current address: Department of Mathematics, Hardin-Simmons University, Abilene, Texas, U.S.A.
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Abstract

If each of E and F is a real Banach space, H a compact Hausdorff space, C(H, E) the Banach space (sup norm ∥·∥∞) of continuous E-valued functions defined on H, L: C(H, E)→F a continuous linear transformation (=operator) with representing measure m, [sum ] the σ-algebra of Borel subsets of H and m˜(A) the semivariation of m on A∈[sum ], then m maps [sum ] into B(E, F**), the Banach space of all operators from E into F** (= the bidual of F), ∥L∥=m˜(H) and L(f)=∫ fdm. The reader may consult [9] or [6] for a detailed discussion of the Riesz representation theorem in this setting.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 124 , Issue 2 , September 1998 , pp. 365 - 369
Copyright
Cambridge Philosophical Society 1998

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