By Lee J.M.

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Furthermore, D |x (f1 , f2 ) · v = ( Df1 |x · v, f2 (x)) + (f1 (x), Df2 |x · v). In particular, if F is an algebra with differentiable product and f 1 : U ⊂ E → F and f2 : U ⊂ E → F then f1 f2 is defined as a function and D(f1 f2 ) · v = (Df1 · v) (f2 ) + (Df1 · v) (Df2 · v). It will be useful to define an integral for maps from an interval [a, b] into a Banach space V. First we define the integral for step functions. A function f on an interval [a, b] is a step function if there is a partition a = t0 < t1 < · · · < tk = b such that f is constant, with value say fi , on each subinterval [ti , ti+1 ).

The set of all maps that are diffeomorphisms near p will be denoted Diff rp (E, F). If f is a C r diffeomorphism near p for all p ∈ U = dom(f ) then we say that f is a local C r diffeomorphism. 3 The space GL(E, F) of continuous linear isomorphisms is an open subset of the Banach space L(E, F). In particular, if id − A < 1 for some N A ∈GL(E) then A−1 = limN →∞ n=0 (id −A)n . 4 The map I :GL(E, F) →GL(E, F) given by taking inverses is a C ∞ map and the derivative of I :g → g −1 at some g0 ∈GL(E, F) is the linear map given by the formula: D I|g0 :A→ −g0−1 Ag0−1 .

In this case, we have for v = (v1 , v2 ), Df (x, y) · (v1 , v2 ) = D1 f (x, y) · v1 + D2 f (x, y) · v2 . The reader will surely not be confused if we also write ∂1 f or ∂x f instead of D1 f (and similarly ∂2 f or ∂y f instead of D2 f ). 1 (Chain Rule) Let U1 and U2 be open subsets of Banach spaces E1 and E2 respectively. Suppose we have continuous maps composing as f g U 1 → U 2 → E3 3 We will often use the letter I to denote a generic (usually open) interval in the real line. 2. CALCULUS ON NORMED SPACES 19 where E3 is a third Banach space.

### Differential and physical geometry by Lee J.M.

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