We establish sufficient conditions for n-fold bounded differentiability (“b-differentiability”) of mappings of locally convex spaces and sufficient conditions for n-fold Hyers-Lang differentiability (“HL-differentiability”) of mappings of pseudotopological linear spaces. We describe a class of locally convex spaces on which there exist everywhere infinitely b-differentiable real functions which are not everywhere continuous (and so are not everywhere HL-differentiable). Our results show, in particular, that for a wide class of locally convex spaces a significant number of the known definitions of C∞-mappings fall into two classes of equivalent definitions.
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Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 729–744 November, 1977.
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Smolyanov, O.G. Higher derivatives of mappings of locally convex spaces. Mathematical Notes of the Academy of Sciences of the USSR 22, 899–906 (1977). https://doi.org/10.1007/BF01098355
- Linear Space
- Real Function
- Wide Class
- High Derivative
- Equivalent Definition