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Generalized L(f) Spaces

Published online by Cambridge University Press:  20 November 2018

D. Hussein ,
M. A. Natsheh  and
I. Qumsiyeh
D. Hussein
Affiliation:
University of Jordan, Amman, Jordan
M. A. Natsheh
Affiliation:
University of Jordan, Amman, Jordan
I. Qumsiyeh
Affiliation:
University of Jordan, Amman, Jordan
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Given any set Γ, let be the family of all finite subsets of . Let f:[0, ∞) → R satisfying: (1) f(x) = 0 if and only if x = 0, (2) f is increasing, (3) f(x + y) ≧ f(x) + f(y) for all x, y ≦ 0, and (4) f is continuous at zero from the right. Such an f is called a modules. Let C be the set of all moduli, and F = {fv ∊ C:v ∊ Γ). Q(Γ) will denote the set of all such F, s. For each F ∊ Q(Γ) let

the summation is taken over Γ, and set

If Γ is countable Q(Γ) will be denoted by Q and LΓ(F) by L(F). Let

Note that

see [4, 5 and 6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 700 - 709
Copyright
Copyright © Canadian Mathematical Society 1985

References

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