New PDF release: A Comprehensive Introduction to Differential Geometry, Vol.

By Michael Spivak

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Vk ) ∈ E1 × · · · × Ek . Now V1 × · · · × Vk is a normed space in several equivalent ways just in the same way that we defined before for the case k = 2. The topology is the product topology. Now one can show with out much trouble that a multilinear map µ : V1 × · · · × Vk → W is bounded if and only if it is continuous. 2. , Ek ; W). , . Then O(E) denotes the group of linear isometries from E onto itself. That is, the bijective linear maps Φ : E→E such that Φv, Φw = v, w for all v, w ∈ E. The group O(E) is called the orthogonal group (or sometimes the Hilbert group in the infinite dimensional case).

Since α is an isomorphism we may compose as follows (α−1 × idE ) ◦ β ◦ A ◦ α−1 ◦ α = (α−1 × idE ) ◦ β ◦ A =δ◦A 24 CHAPTER 1. BACKGROUND to get a map that is easily seen to have the correct form. If A is a splitting injection as above it easy to see that there are closed subspaces F1 and F2 of F such that F = F1 ⊕ F2 and such that A maps E isomorphically onto F1 . 13 Let A : E → F be an surjective continuous linear map. We say that A is a splitting surjection if there are Banach spaces E1 and E2 with E∼ = E1 × E2 and if A is equivalent to the projection pr1 : (x, y) → x.

8 (Implicit Function Theorem II) Let E1 , E2 and F be Banach spaces and U × V ⊂ E1 × E2 open. Let f : U × V → F be a C r mapping such that f (x0 , y0 ) = w0 . If D2 f (x0 , y0 ) : E2 → F is a continuous linear isomorphism then there exists (possibly smaller) open sets U0 ⊂ U and W0 ⊂ F with x0 ∈ U0 and w0 ∈ W0 together with a unique mapping g : U0 × W0 → V such that f (x, g(x, w)) = w for all x ∈ U0 . Here unique means that any other such function h defined on a neighborhood U0 × W0 will equal g on some neighborhood of (x0 , w0 ).

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