By Nicolas Bourbaki
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Additional resources for Elements of Mathematics: General Topology, Pt.2
Let X and Y be two metric spaces. A sequence of maps (fn )n≥0 from X to Y is said to be uniformly equicontinuous 1 if for every > 0, there exists an η > 0 such that for every integer n ≥ 0 and for every x and y in X, we have |x − y| < η ⇒ |fn (x) − fn (y)| < . An example of a uniformly equicontinuous sequence is a sequence (fn )n≥0 of K-Lipschitz maps, with K independent of n or, more generally, a sequence of uniformly Hölder maps. More generally, if there exist two constants α > 0 and K ≥ 0 such that for any integer n ≥ 0 and for all x and y in X we have |fn (x) − fn (y)| ≤ K|x − y|α , then the sequence (fn ) is uniformly equicontinuous.
If γ : [0, 1] → X is a constant path, then the conclusion follows trivially. Now suppose that γ = γ ψ, where γ : [c, d] → X is a path that is parametrized by arclength and where ψ : [0, 1] → [c, d] is defined by ψ(x) = (d − c)x + c. 9. 3 Differentiable paths in Euclidean space For all n ≥ 1, we denote by E = En the Euclidean space of dimension n, that is, the space Rn equipped with the norm (x1 , . . , xn ) = metric induced by that norm. 1. Let γ : [a, b] → E be a C 1 -path and let γ : [a, b] → E be its derivative.
For all n ≥ 0 and for all t in [0, 1], let γn : [0, 1] → E2 be the map defined by γn (t) = (t, Fn (t)). 3 the images of the paths γ1 , γ2 , γ3 and γ4 . 3. 1 (i). The sequence of paths (γn ) converges to the path √ γ : [0, 1] → E2 defined by γ (t) = (t, 0), whose length is 1, whereas L(γn ) = 2 for every n ≥ 0. Thus, L(γn ) → L(γ ). (ii) Let γn : [0, π] → R be the path defined by γn (t) = (1/n) cos(n2 t). When t varies in the interval [0, π], cos(n2 t) takes n2 times the values 1 and −1. Hence, 1 L(γn ) ≥ × 2n2 = 2n.
Elements of Mathematics: General Topology, Pt.2 by Nicolas Bourbaki