One can ask whether a nonlinear map between locally convex topological vector spaces is smooth according to various definitions. The definition due to Michal, developed by Bastiani, is a very general such notion.
A map $\Phi:E\rightarrow F$ from a locally convex topological vector space (lctvs) $E$ into another lctvs $F$ is said to be Michal-Bastiani smooth (or MB-smooth) if its directional derivatives of order $k$,
exist and the maps $D^k\Phi:E\times E^k\rightarrow F$ are jointly continuous? for all $k\in\mathbb{N}$. The notion is due to Michal (Michal 38,Michal 40), and was further developed by (Bastiani 64).
This notion of smoothness is the one used in Milnor‘s treatment of infinite-dimensional Lie groups (Milnor 84) and Hamilton’s exposé of the Nash-Moser inverse function theorem (Hamilton 92). A comparison to other notions of smoothness for topological vector spaces with various properties is in (Keller 74).
MB-smooth maps have the following properties.
Continuous linear maps are MB-smooth;
MB-smooth maps are continuous;
MB-smooth maps are smooth in the sense of convenient vector spaces, since the chain rule holds;
Maps between Fréchet spaces are MB-smooth if and only if they are conveniently smooth.
Note that a map that is smooth in the sense of a convenient vector space (equivalently, a smooth map between the corresponding diffeological spaces) is not necessarily continuous, so not all convenient smooth maps between lctvs are MB-smooth. Explicit examples have been given in (Glöckner 06). It is even the case that one can have a convenient isomorphism that is not MB-smooth, so the faithful and non-full functor
is not even injective on isomorphism classes. The following example was supplied in (TaQ 15):
Let $E$ be the vector space of all real (two-sided) sequences $x=\langle\sp x_i\mid i\in\mathbb{Z}\rangle$ for which $x_i=0$ for $i\ll 0$, topologized so that we get a linear homeomorphism $E\to\mathbb{R}^\mathbb{N}\times\mathbb{R}^{(\mathbb{N})}$ defined by $x\mapsto(u,v)$ where $u_i=x_{i-1}$ and $v_i=x_{-i}$ for i\gt 0}
.
Then define the bijection $i\gt 0}f:E\to E$ by $x\mapsto y$ where $y_i=x_i$ for $i\neq 0$ and $y_0=x_0+\sum_{i\in\mathbb{N}}(x_i\cdot x_{-i})$. The inverse is given by $y\mapsto x$ where $y_i=x_i$ for $i\neq 0$ and $x_0=y_0-\sum_{i\in\mathbb{N}}(y_i\cdot y_{-i})$. The duality map $\mathbb{R}^\mathbb{N}\times\mathbb{R}^{(\mathbb{N})}\to\mathbb{R}$, $x\mapsto \sum_{i\in\mathbb{N}}(x_i\cdot x_{-i})$, is discontinuous but bornological, and hence a conveniently smooth bilinear map. Thus $f$ is discontinuous and so not MB-smooth.
Aristotle Demetrius Michal, Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. USA 24 (1938), 340-342 (pdf)
Aristotle Demetrius Michal, Differential of functions with arguments and values in topological abelian groups, Proc. Nat. Acad. Sci. USA 26 (1940), 356–359. (pdf)
Andree Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J. Anal. Math. 13 (1964), 1–114. (article, paywalled)
John Milnor, Remarks on Infinite-Dimensional Lie Groups, in: B. DeWitt, R. Stora, eds., Les Houches Session XL, Relativity, Groups and Topology II (North-Holland, 1984), pp. 1007-1057
Richard Hamilton, The Inverse Function Theorem of Nash and Moser. Bull. Amer. Math. Soc. 7 (1982) 65-222.
Keller, H. H., Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics 417, Springer-Verlag, 1974 (Springerlink, paywalled)
Helge Glöckner?, Discontinuous non-linear mappings on locally convex direct limits, Publ. Math. Debrecen 68 (2006) 1-13, arXiv:math/0503387.
TaQ (user 12643), Answer to Do locally convex topological vector spaces embed into diffeological spaces?, MathOverflow, <http://mathoverflow.net/q/209470> (version: 2015-06-16)
Last revised on June 16, 2015 at 23:08:42. See the history of this page for a list of all contributions to it.