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By Patrick Iglesias-zemmour

Diffeology is the 1st textbook at the topic. it truly is aimed to graduate scholars and researchers  who paintings in differential geometry or in mathematical physics

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Let X and X be two diffeological spaces, whose diffeologies are denoted by D and D . The notion of pushforward of diffeologies gives a new interpretation of the notion of differentiability: a map SUBDUCTIONS 25 f : X → X is smooth if and only if f∗ (D) ⊂ D . In other words, C∞ (X, X ) = {f ∈ Maps(X, X ) | f∗ (D) ⊂ D }. Proof. Let us assume first that f is smooth. Said differently, D is a diffeology of X such that f is smooth. But f∗ (D) is the intersection of all the diffeologies of X for which f is smooth (art.

Dxk dxk ˆ : f → f(x), from C∞ (R) to R, is 2) Show that, for every real number x, the map x smooth. Deduce that, for every real number x, for every integer k, the following map, called the k-jet or the jet of order k, is smooth: Dkx : C∞ (R) → Rk+1 defined by Dkx (f) = (f(x), f (x), . . , f(k) (x)). 3) Show that, for any pair of real numbers a and b, the following map Ia,b is smooth, where the sign denotes the Riemann integral: Ia,b : C∞ (R) → R b with Ia,b (f) = f(t) dt . a 4) Let C∞ 0 (R) be the space of smooth real maps f such that f(0) = 0.

3) Generalize this construction to C∞ (Rn , Rn ). What can you say about the injection of GL(n, R) into C∞ (Rn )? SUMS OF DIFFEOLOGICAL SPACES 21 Sums of Diffeological Spaces The category {Diffeology} is closed for coproducts (also called sums or disjoint unions) of diffeological spaces. There exists a distinguished diffeology, called the sum diffeology, on the sum of any family of diffeological spaces. Reminder. Let us recall the formal construction of the sum (also called coproduct) of any family of sets [Bou72].

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