Weak-star convergence in the dual of the continuous functions on the -cube,

Authors:
Richard B. Darst and Zorabi Honargohar

Journal:
Trans. Amer. Math. Soc. **275** (1983), 357-372

MSC:
Primary 46E27; Secondary 26B30, 28A33, 60B10

MathSciNet review:
678356

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a positive integer and let denote the -cube. Let denote the (sup norm) space of continuous (real-valued) functions defined on , and let denote the (variation norm) space of (real-valued) signed Borel measures defined on the Borel subsets of . Let be a sequence of elements of . Necessary and sufficient conditions are given in order that exists for every . After considering a finite dimensional case, the infinite dimensional case is entertained.

**[1]**T. M. Apostol,*Mathematical analysis*, Addison-Welsey, Reading, Mass., 1957.**[2]**George Bachman and Lawrence Narici,*Functional analysis*, Academic Press, New York-London, 1966. MR**0217549****[3]**Paul R. Halmos,*Measure Theory*, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR**0033869****[4]**Göran Högnäs,*Characterization of weak convergence of signed measures on [0,1]*, Math. Scand.**41**(1977), no. 1, 175–184. MR**0482909****[5]**T. H. Hildebrandt,*Introduction to the theory of integration*, Pure and Applied Mathematics, Vol. XIII, Academic Press, New York-London, 1963. MR**0154957****[6]**James R. Munkres,*Topology: a first course*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR**0464128****[7]**I. P. Natanson,*Theory of functions of a real variable*, Vol. I, 3rd printing, Ungar, New York, 1964.**[8]**H. L. Royden,*Real analysis*, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR**0151555****[9]**Walter Rudin,*Real and complex analysis*, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR**0344043****[10]**Walter Rudin,*Principles of mathematical analysis*, Second edition, McGraw-Hill Book Co., New York, 1964. MR**0166310**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46E27,
26B30,
28A33,
60B10

Retrieve articles in all journals with MSC: 46E27, 26B30, 28A33, 60B10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1983-0678356-9

Article copyright:
© Copyright 1983
American Mathematical Society