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These are the notion of equicontinuity and the Ascoli theorem. Equicontinuity Let (X, d) and (Y, σ) be metric spaces and let F = {ϕ : X → Y } be a family of functions. We call F equicontinuous at x0 ∈ X if for every ε > 0 there is δ > 0 such that d(x, x0 ) < δ implies σ(ϕ(x), ϕ(x0 )) < ε for all ϕ ∈ F. We say that F is equicontinuous if it is equicontinuous at each x0 ∈ X. Next, the central result linking normal families and equicontinuity is the Arzela-Ascoli theorem. 9. If (X, d) is a separable metric space and (Y, σ) a compact metric space, then every equicontinuous family of mappings F = {ϕ : X → Y } is a normal family.

4719 . .. 10. 1. An elementary proof using classical complex analysis can be found in [6]. 15. If f is holomorphic in D, if it is continuous in D and if f ∂D is homotopic to the identity relative to C \ {0}, then f (z0 ) = 0 at precisely one point z0 ∈ D. 10 Distortion by Conformal Mapping From a historical perspective, the notion of a quasiconformal mapping has its roots in the study of the geometric properties of conformal mappings. The philosophy here is that locally a conformal mapping is close to its linearization z → f (z0 ) + f (z0 )(z − z0 )—a similarity mapping of C.

CHAPTER 2. CONFORMAL GEOMETRY 18 Reflecting this phenomenon, we shall show that in fact the space S(2) is isometric to the hyperbolic plane D. 16) In fact, this correspondence goes much deeper than just this one-to-one isomorphism, and as we shall see it partly reflects the correspondence between the real and complex distortion functions of quasiconformal mappings. We shall see it appearing in a number of different forms. The space S(2) is clearly two-dimensional, and any A ∈ S(2) can be written in the form λ 0 A = Ot O 0 1/λ for some O ∈ SO(2, R) and λ > 0.

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