## Weak compactness in locally convex spaces

HTML articles powered by AMS MathViewer

- by D. G. Tacon PDF
- Trans. Amer. Math. Soc.
**180**(1973), 463-474 Request permission

## Abstract:

The notion of weak compactness plays a central role in the theory of locally convex topological vector spaces. However, in the statement of many theorems, completeness of the space, or at least quasi-completeness of the space in the Mackey topology is an important assumption. In this paper we extend the concept of weak compactness in a general way and obtain a number of useful particular cases. If we replace weak compactness by these generalized notions we can drop the completeness assumption from the statement of many theorems; for example, we generalize the classical theorems of Eberlein and Kreĭn. We then consider generalizations of semireflexivity and reflexivity and characterize these properties in terms of our previous ideas as well as in terms of known concepts. In most of the proofs we use techniques of nonstandard analysis.## References

- Mahlon M. Day,
*Normed linear spaces*, Reihe: Reelle Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR**0094675** - R. J. Fleming,
*Characterizations of semi-reflexivity and quasi-reflexivity*, Duke Math. J.**36**(1969), 73–80. MR**236646**, DOI 10.1215/S0012-7094-69-03611-4 - A. Grothendieck,
*Critères de compacité dans les espaces fonctionnels généraux*, Amer. J. Math.**74**(1952), 168–186 (French). MR**47313**, DOI 10.2307/2372076 - Robert C. James,
*Weak compactness and reflexivity*, Israel J. Math.**2**(1964), 101–119. MR**176310**, DOI 10.1007/BF02759950 - J. L. Kelley and Isaac Namioka,
*Linear topological spaces*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR**0166578**, DOI 10.1007/978-3-662-41914-4
G. Köthe, - Elton Lacey and R. J. Whitley,
*Conditions under which all the bounded linear maps are compact*, Math. Ann.**158**(1965), 1–5. MR**173159**, DOI 10.1007/BF01370391 - W. A. J. Luxemburg,
*A general theory of monads*, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR**0244931** - Abraham Robinson,
*Non-standard analysis*, North-Holland Publishing Co., Amsterdam, 1966. MR**0205854** - Helmut H. Schaefer,
*Topological vector spaces*, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR**0193469** - D. G. Tacon,
*Weak compactness in normed linear spaces*, J. Austral. Math. Soc.**14**(1972), 9–16. MR**0315405**, DOI 10.1017/S1446788700009563 - Albert Wilansky,
*Functional analysis*, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR**0170186**

*Topologische linear Räume*. I, Die Grundlehren der math. Wissenschaften, Band 107, Springer-Verlag, Berlin, 1960; English transl., Die Grundlehren der math. Wissenschaften, Band 159, Springer-Verlag, New York, 1969. MR

**24**#A411; MR

**40**#1750.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**180**(1973), 463-474 - MSC: Primary 46A25
- DOI: https://doi.org/10.1090/S0002-9947-1973-0322467-8
- MathSciNet review: 0322467