nLab Michal-Bastiani smooth map

Idea

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.

Definition

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 $k$,

$$D^k\Phi[x](y_1,\ldots,y_k) = \left.\frac{\partial^k}{\partial\lambda_1\cdots\partial\lambda_k}\right|_{\lambda_1=\cdots=\lambda_k=0}\Phi\left(x+\sum^k_{j=1}\lambda_j y_j\right)$$

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).

Properties

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

$$lctvs_{MB} \to lctvs_{convenient} \hookrightarrow DiffeologicalSpace$$

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.

References

• 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)