Smooth locally convex spaces

Lloyd, John

doi:10.1090/S0002-9947-1975-0380868-8

Smooth locally convex spaces
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by John Lloyd

Trans. Amer. Math. Soc. 212 (1975), 383-392

DOI: https://doi.org/10.1090/S0002-9947-1975-0380868-8

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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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