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

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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.

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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