0
$\begingroup$

Naively, one could topologise the set of smooth (ie $\mathcal{C}^\infty$) maps between two smooth manifolds $M \to N$ with the subspace topology $\mathcal{C}^\infty(M,N) \subseteq \mathcal{C}^0(M,N)$, where the usual compact-open topology is considered in $\mathcal{C}^0(M,N)$.

Now, why are the weak/strong topologies in $\mathcal{C}^\infty(M,N)$ considered instead? How different are they from the subspace topology as explained before? What are their advantages with respect to the subspace topology?

Improve this question
$\endgroup$
3
  • $\begingroup$ Peter Michor's "Convenient setting of global analysis" is a very densely written book, but answers your question in great generality. $\endgroup$
    – Denis T
    Commented Mar 2, 2022 at 14:59
  • $\begingroup$ Also very large portion of @AndrewStacey works are aimed at studying function spaces of smooth manifolds. $\endgroup$
    – Denis T
    Commented Mar 2, 2022 at 15:02
  • $\begingroup$ The book my Kriegl & Michor is of course very comprehensive, but a more down-to-earth reference is "Differential topology" by Hirsch (Springer, 1976). $\endgroup$
    – Igor Khavkine
    Commented Mar 2, 2022 at 15:51

0

Reset to default

You must log in to answer this question.