By Lee J.M.

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Y). With this notation we have the following version of Taylor’s theorem. 5 (Taylor’s theorem) Given Banach spaces V and W,a C r function f : U → W and a line segment t → (1 − t)x + ty contained in U, we have that t → D p f (x + ty) · (y)p is defined and continuous for 1 ≤ p ≤ k and f (x + y) = f (x) + 1 + 0 1 2 1 Df |x · y + D f 1! 2! x · (y)2 + · · · + (1 − t)k−1 k D f (x + ty) · (y)k dt (k − 1)! 1 Dk−1 f (k − 1)! x · (y)k−1 The proof is by induction and follows the usual proof closely. The point is that we still have an integration by parts formula coming from the product rule and we still have the fundamental theorem of calculus.

The objects of this category are pairs (U, p) where U ⊂ E a Banach space and p ∈ E. The morphisms are maps of the form f : (U, p) → (V, q) where f (p) = q. As before, we shall let f :: (E, p)→(F, q) refer to such a map in case we do not wish to specify the opens sets U and V . Let X, Y be topological spaces. When we write f :: X → Y we imply only that f is defined on some open set in X. We shall employ a similar use of the symbol “ :: ” when talking about continuous maps between (open subsets of) topological spaces in general.

We form another map by composition of isomorphisms; β −1 ◦ β ◦ A ◦ α−1 ◦ (β, idE2 ) = A ◦ α−1 ◦ (β, idE2 ) := A ◦ δ and check that this does the job. If A is a splitting surjection as above it easy to see that there are closed subspaces E1 and E2 of E such that E = E1 ⊕ E2 and such that A maps E onto E1 as a projection x + y → x. Local (nonlinear) case. For the nonlinear case it is convenient to keep track of points. More precisely, we will be working in the category of local pointed maps. The objects of this category are pairs (U, p) where U ⊂ E a Banach space and p ∈ E.