Geodesic
Higher derivatives of mappings of locally convex spaces
Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 729-744

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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 $C^\infty$-mappings fall into two classes of equivalent definitions. 
@article{MZM_1977_22_5_a11,
     author = {O. G. Smolyanov},
     title = {Higher derivatives of mappings of locally convex spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {729--744},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a11/}
}
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O. G. Smolyanov. Higher derivatives of mappings of locally convex spaces. Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 729-744. http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a11/