The approximation of linear operators

Authors:
J. W. Brace and P. J. Richetta

Journal:
Trans. Amer. Math. Soc. **157** (1971), 1-21

MSC:
Primary 47.55

DOI:
https://doi.org/10.1090/S0002-9947-1971-0278122-4

MathSciNet review:
0278122

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the vector space of all linear maps of into . Consider a subspace of such as all continuous maps. In distinguish a subspace of maps which are to be approximated by members of a smaller subspace of . Thus we always have . Then the approximation problem which we consider is to find a locally convex linear Hausdorff topology on such that or the completion of is .

In the case where and are Banach spaces, we have approximation topologies for (i) all linear operators, (ii) all the continuous linear operators, (iii) all weakly compact operators, (iv) all completely continuous operators, (v) all compact operators, and (vi) certain subclasses of the strictly singular operators.

Our method is that of considering members of as linear forms on . Each class of linear operators is characterized as a family of linear forms. We exploit these characterizations to develop the needed topologies. Convergence on filters appears as a natural tool in doing this; indeed, in the case of linear forms we can obtain every relevant topology via convergence on filters. Particular examples give representations of weak topologies. A by-product of the main results is that Grothendieck's approximation condition holds when we have the weak topology on a locally convex space.

**[1]**John W. Brace,*The topology of almost uniform convergence*, Pacific J. Math.**9**(1959), 643–652. MR**0109293****[2]**John W. Brace,*Approximating compact and weakly compact operators*, Proc. Amer. Math. Soc.**12**(1961), 392–393. MR**0130575**, https://doi.org/10.1090/S0002-9939-1961-0130575-7**[3]**John W. Brace,*Convergence on filters and simple equicontinuity*, Illinois J. Math.**9**(1965), 286–296. MR**0176438****[4]**John W. Brace and Robert M. Nielsen,*A uniform boundedness theorem*, Proc. Amer. Math. Soc.**18**(1967), 624–627. MR**0212532**, https://doi.org/10.1090/S0002-9939-1967-0212532-0**[5]**J. W. Brace, G. D. Friend, and P. J. Richetta,*Locally convex topologies on function spaces*, Duke Math. J.**36**(1969), 709–714. MR**0253008****[6]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[7]**Alexandre Grothendieck,*Produits tensoriels topologiques et espaces nucléaires*, Mem. Amer. Math. Soc.**No. 16**(1955), 140 (French). MR**0075539****[8]**Richard H. Herman,*Operator representation theorems*, Proc. Amer. Math. Soc.**19**(1968), 372–376. MR**0222705**, https://doi.org/10.1090/S0002-9939-1968-0222705-X**[9]**J. Horváth,*Topological vector spaces and distributions*. Vol. 1, Addison-Wesley, Reading, Mass., 1966. MR**34**#4863.**[10]**John L. Kelley,*General topology*, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR**0070144****[11]**Irving Kaplansky,*Functional analysis*, Some aspects of analysis and probability, Surveys in Applied Mathematics. Vol. 4, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958, pp. 1–34. MR**0101475****[12]**Ivan Singer,*On the basis problem in topological linear spaces*, Rev. Roumaine Math. Pures Appl.**10**(1965), 453–457. MR**0194875****[13]**J. W. Brace,*The space of continuous linear operators as a completion of 𝐸′⊗𝐹*, Bull. Amer. Math. Soc.**75**(1969), 821–823. MR**0246087**, https://doi.org/10.1090/S0002-9904-1969-12310-4

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47.55

Retrieve articles in all journals with MSC: 47.55

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0278122-4

Keywords:
Approximation of linear operators,
compact operators,
weakly compact operators,
completely continuous operators,
tensor products,
convergence on filters,
completions,
Grothendieck's completion theorem,
Grothendieck's approximation property,
weak topologies

Article copyright:
© Copyright 1971
American Mathematical Society