Abstract
We show that when E ⊂ ℝn is a compact Whitney p-regular domain then the usual and quotient topologies for C∞ (E) are equivalent with a special estimate of the continuity constants. It follows that in the equivalence of Markov and Sobolev type inequalities given in [2], the quotient norm may be replaced by the usual norm in case E is Whitney p-regular.
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References
Bierstone, E. (1980) ‘Differentiable Functions’, Bol. Soc. Bras. Mat., vol. 11, no. 2, 139–190.
Bos, L. and Milman P. (1991) ‘The equivalence of Markov and Sobolev type inequalities on compact subsets inℝn’, preprint.
Tougeron, J. (1972) Idéaux de fonctions différentiables, Springer, Berlin.
Whitney, H. (1934) ‘Analytic extensions of differentiable functions defined in closed sets’, Trans. Amer. Math. Soc. 36, 369–387.
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© 1992 Springer Science+Business Media Dordrecht
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Bos, L.P., Milman, P.D. (1992). The Equivalence of the Usual and Quotient Topologies for C ∞(E) when E ⊂ ∝n is Whitney p-Regular. In: Singh, S.P. (eds) Approximation Theory, Spline Functions and Applications. NATO ASI Series, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2634-2_15
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DOI: https://doi.org/10.1007/978-94-011-2634-2_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5164-4
Online ISBN: 978-94-011-2634-2
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