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I'm planning a short course on few topics and applications of nonlinear functional analysis, and I'd like a reference for a quick and possibly self-contained construction of a structure of a Banach differentiable manifold for the space of continuous mappings $C^0(K,M)$, where $K$ is a compact topological space (even metric if it helps) and $M$ is a (finite dimensional) differentiable manifold.

A construction of a differentiable structure of Banach manifold for this space can be found e.g. in Lang's book Fundamentals of differential geometry (1999). The main tools are the exponential map and tubular nbds (having fixed a Riemannian structure on $M$. This is OK but I believe there should be something even more basic.

Does anybody have a reference for alternative constructions (not necessarily elementary) ?

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2 Answers 2

up vote 6 down vote accepted

Well, there is always my old "Foundations of Global Nonlinear Analysis", which is about just this.

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Are there copies of this book available? Doing a little Googling around I wasn't able to find it. –  Ryan Budney Sep 3 '10 at 1:19

I tend to focus more on the manifold of smooth maps, so this list should be understood as being more from that perspective than of continuous maps, but there's a fair amount of overlap in the two cases so I hope it's of some use.

  1. Constructing Smooth Manifolds of Loop Spaces. Despite my disqualifier above, this does deal with continuous maps as well as smooth ones. However, it deals with $K = S^1$ only. On the other hand, the role of $K$ in the construction is "not a lot" so understanding the case $K = S^1$ gives a lot of insight about the general case.

  2. Manifolds of differentiable mappings by Peter Michor. Actually, search MathSciNet for Michor with "manifold" in the title.

  3. The Convenient Setting of Global Analysis contains a whole section on manifolds of mapping spaces.

  4. nLab pages: Manifold structure of mapping spaces. Bit of a "work in progress" and concentrating on smooth maps; the bit that might interest you is the condition on the source: all that is needed is that it be sequentially compact.

  5. The differential topology of loop spaces. An interpolation between (1) and (3) above. Written for people without the stamina to read (3) but who were interested in loop spaces. Again, concentrates on "smooth", but as (1) shows then that's not really important.

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Thank you, I'll have a good reading! I think I once had a glance to 3. Isn't that an important feature of these works is the differential of $f:U\subset X\to Y$ as a map $Df:U\times X\to Y$ rather than a map $Df:U\to L(X,Y)$? (which in infinite dimension really makes a different notion of a continuously differentiable map)? –  Pietro Majer Sep 2 '10 at 15:25
    
The great thing about the calculus of Kriegl, Michor, Frolicher (and others) is that continuity is secondary. So worrying about whether or not a map is continuous or what topology to put on L(X,Y) is no longer relevant. What matters is the effect on smooth curves, and in that realm then the exponential law holds and the derivative can be viewed in both ways. –  Loop Space Sep 2 '10 at 18:58

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