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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Smooth locally convex spaces


Author: John Lloyd
Journal: Trans. Amer. Math. Soc. 212 (1975), 383-392
MSC: Primary 58C20; Secondary 46A05
DOI: https://doi.org/10.1090/S0002-9947-1975-0380868-8
MathSciNet review: 0380868
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Abstract: The main theorem is

Let E be a separable (real) Fréchet space with a nonseparable strong dual. Then E is not strongly $ D_F^1$-smooth.

It follows that if X is uncountable, locally compact, $ \sigma $-compact, metric space, then $ C(X)$ (with the topology of compact convergence) does not have a class of seminorms which generate its topology and are Fréchet differentiable (away from their null-spaces).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0380868-8
Keywords: Smooth locally convex space, locally convex direct sum, Fréchet derivative, Hadamard derivative
Article copyright: © Copyright 1975 American Mathematical Society