|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
can someone please explain, what is 'an r-times continuously differentiable function'? How is differentiability for mappings between banach spaces defined? or is this somehow inplicit in the abstract tangent space notion?
- It's a little implicit, but not too abstract, and is formally just the same as Fréchet differentiability for maps between finite-dimensional real spaces. A map f : U → Y defined on an open subset U of a Banach space X and taking values in a second Banach space Y is differentiable if, for each point x of U, there is a linear map (the derivative (df)(x)) from TxU = X into Y such that [insert your favourite version of Fréchet derivative here]. Put another way, the derivative df is a function U → Lin(X; Y). But Lin(X; Y) is itself a Banach space, so if df : U → Lin(X; Y) is smooth enough, we can rinse and repeat to get a second derivative d2f : U → Lin(X; Lin(X; Y)). Up to isomorphism, Lin(X; Lin(X; Y)) is the same as Lin(X × X; Y). In other words, at each point x of U, (d2f)(x) is a bilinear map, one that "eats" two tangent directions to U at x and "spits out" a tangent vector to f(U) at f(x). Continuing in this fashion, the rth derivative drf is a map from U into Lin(Xr; Y). Just as in elementary analysis, an r-times continuously differentiable function is one for which drf exists and is a continuous function. Sullivan.t.j 00:56, 2 July 2007 (UTC)
A Global Chart ?
I would like to add an important fact into this article, but I'm not sure if this is really true: "Every infinite dimensional Banach manifold admits a global chart, i.e. every infinite dimensional Banach manifold is diffeomorphic to a subset of Banach space."
Does anyone know if this statement is correct? If so, it would a good thing to add it because it points out one main difference between the (inf.dim.) Banach case and the ordinary finite dimensional case. (in finite dimensions it remains true that there is always a finite atlas - which is a surprising fact for its own) --188.8.131.52 (talk) 11:59, 16 April 2008 (UTC)
- This is an interesting question, but I think that it first needs a little refinement: is it true that every infinite-dimensional Banach manifold, M, is diffeomorphic to some open subset of some Banach space, Y? Sullivan.t.j (talk) 15:17, 16 April 2008 (UTC)
Math priority rating
I think that this concept should receive at least a "mid importance rating" in comparison to other articles since "low importance" is simply too low in my opinion. Any comments? --PST 07:52, 9 September 2009 (UTC)