Strong convergence of functions on Köthe spaces
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- by Gerald Silverman PDF
- Trans. Amer. Math. Soc. 165 (1972), 27-35 Request permission
Abstract:
Let $\Lambda$ be a rearrangement invariant Köthe space over a nondiscrete group G with Haar measure $\mu$. For a function $f \in \Lambda$ and relatively compact 0-neighborhood U in G the function \[ {T_U}f(x) = \frac {1}{{\mu (U)}} \cdot \int _{U + x} {f d\mu } \] is continuous and also belongs to $\Lambda$. The convergence ${T_U}f \to f$ (as $U \to 0$) for the strong Köthe topology on $\Lambda$ is involved in establishing compactness criteria for subsets of a Köthe space. The main result of this paper is a necessary and sufficient condition for convergence ${T_U}f \to f$ in the strong topology on $\Lambda$.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 165 (1972), 27-35
- MSC: Primary 46A45; Secondary 46E30
- DOI: https://doi.org/10.1090/S0002-9947-1972-0291790-7
- MathSciNet review: 0291790