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\begin{document}

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\section*{Michal-Bastiani smooth map}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 [[Andree Ehresmann|Bastiani]], is a very general such notion.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A map $\Phi:E\rightarrow F$ from a [[locally convex topological vector space]] (lctvs) $E$ into another lctvs $F$ is said to be \textbf{Michal-Bastiani smooth} (or \textbf{MB-smooth}) if its [[directional derivatives]] of order $k$,

\begin{displaymath}
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)
\end{displaymath}
exist and the maps $D^k\Phi:E\times E^k\rightarrow F$ are [[jointly continuous map|jointly continuous]] for all $k\in\mathbb{N}$. The notion is due to Michal (\hyperlink{Michal38}{Michal 38},\hyperlink{Michal40}{Michal 40}), and was further developed by (\hyperlink{Bastiani64}{Bastiani 64}).

This notion of smoothness is the one used in [[Milnor]]`s treatment of infinite-dimensional [[Lie groups]] (\hyperlink{Milnor}{Milnor 84}) and Hamilton's expos\'e{} of the Nash-Moser [[inverse function theorem]] (\hyperlink{Hamilton}{Hamilton 92}). A comparison to other notions of smoothness for topological vector spaces with various properties is in (\hyperlink{Keller}{Keller 74}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

MB-smooth maps have the following properties.

\begin{itemize}%
\item Continuous [[linear maps]] are MB-smooth;


\item MB-smooth maps are [[continuous map|continuous]];


\item MB-smooth maps are smooth in the sense of [[convenient vector spaces]], since the [[chain rule]] holds;


\item Maps between [[Fréchet spaces]] are MB-smooth if and only if they are conveniently smooth.



\end{itemize}
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 (\hyperlink{Glockner06}{Gl\"o{}ckner 06}). It is even the case that one can have a convenient \emph{isomorphism} that is not MB-smooth, so the faithful and non-full functor

\begin{displaymath}
lctvs_{MB} \to lctvs_{convenient} \hookrightarrow DiffeologicalSpace
\end{displaymath}
is not even injective on isomorphism classes. The following example was supplied in (\hyperlink{TaQ15}{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 $f:E\to E$ 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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Aristotle Demetrius Michal, \emph{Differential calculus in linear topological spaces}, Proc. Nat. Acad. Sci. USA \textbf{24} (1938), 340-342 (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077109/pdf/pnas01796-0040.pdf}{pdf})


\item Aristotle Demetrius Michal, \emph{Differential of functions with arguments and values in topological abelian groups}, Proc. Nat. Acad. Sci. USA \textbf{26} (1940), 356--359. (\href{http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078188/pdf/pnas01616-0038.pdf}{pdf})


\item [[Andree Bastiani]], \emph{Applications diff\'e{}rentiables et vari\'e{}t\'e{}s diff\'e{}rentiables de dimension infinie}, J. Anal. Math. \textbf{13} (1964), 1--114. (\href{http://dx.doi.org/10.1007/BF02786619}{article}, paywalled)


\item [[John Milnor]], \emph{Remarks on Infinite-Dimensional Lie Groups}, in: B. DeWitt, R. Stora, eds., Les Houches Session XL, \emph{Relativity, Groups and Topology II} (North-Holland, 1984), pp. 1007-1057


\item [[Richard Hamilton]], \emph{The Inverse Function Theorem of Nash and Moser}. Bull. Amer. Math. Soc. \textbf{7} (1982) 65-222.


\item Keller, H. H., \emph{Differential Calculus in Locally Convex Spaces}, Lecture Notes in Mathematics 417, Springer-Verlag, 1974 (\href{http://dx.doi.org/10.1007/BFb0070564}{Springerlink}, paywalled)


\item [[Helge Glöckner]], \emph{Discontinuous non-linear mappings on locally convex direct limits}, Publ. Math. Debrecen 68 (2006) 1-13, \href{http://arxiv.org/abs/math/0503387}{arXiv:math/0503387}.


\item TaQ (\href{http://mathoverflow.net/users/12643/taq}{user 12643}), Answer to \emph{Do locally convex topological vector spaces embed into diffeological spaces?}, MathOverflow, $<$http://mathoverflow.net/q/209470{\tt \symbol{62}} (version: 2015-06-16)



\end{itemize}
[[!redirects Michal-Bastiani smooth maps]] [[!redirects MB-smooth maps]] [[!redirects MB-smooth map]]



\end{document}