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A note on Köthe spaces

Published online by Cambridge University Press:  18 May 2009

Nguyen Phuong Các
Nguyen Phuong Các
Affiliation:
University of Iowa, Iowa City, Iowa 52240, U.S.A.
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Let E be a locally compact space which can be expressed as the union of an increasing sequence of compact subsets Kn (n =1, 2, …) and let μ be a positive Radon measure on E. Ω is the space of equivalence classes of locally integrable functions on E. We denote the equivalence class of a function f by and if is an equivalence class then f denotes any function belonging to f. Provided with the topology defined by the sequence of seminorms

Ω is a Fréchet space. The dual of Ω is the space φ of equivalence classes of measurable, p.p. bounded functions vanishing outside a compact subset of E. For a subset Γ of Ω, the collection Λ of all ∊Ω, such that for each g∊Γ the product fg is integrable, is called a Köthe space and Γ is said to be the denning set of Λ. The Köthe space Λx which has Λ as a denning set is called the associated Kothe space of Λ. Λ and Λx are put into duality by the bilinear form

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 11 , Issue 2 , July 1970 , pp. 152 - 155
Copyright
Copyright © Glasgow Mathematical Journal Trust 1970

References

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