Metrics of curves in shape optimization and analysis - Andrea Carlo ...

shapes. Unfortunately, H 0 does not yield a well define metric structure, sincethe associated distance is identically zero.So to achieve our goal, we will need to devise new metrics.3 Basic mathematical notionsIn this section we provide the mathematical theory that will be needed in therest of the course. (Some of the definitions are usually known to mathematics’students; we will present them nonetheless as a chance to remark less knownfacts.)We will though avoid technicalities in the definitions, and for the most partjust provide a base intuition of the concepts. The interested reader may obtainmore details from a books in analysis, such as [2], in functional analysis, suchas [48], and in differential and Riemannian geometry, such as [14], [29] or [30].We start with a basic notion, in a quite simplified form.Definition 3.1 (Topological spaces) A topological space is a set M withassociated a topology τ of subsets, that are the open sets in M.The topology is the simplest and most general way to define what are the“convergent sequences of points” and the “continuous functions”. We will notprovide details regarding topological spaces, since in the following we will mostlydeal with normed spaces and metric spaces, where the topology is induced by anorm or a metric. We just recall this definition.Definition 3.2 A homeomorphism is an invertible continuous function φ :M → N between two topological spaces M, N, whose inverse φ −1 is againcontinuous.3.1 Distance and metric spaceDefinition 3.3 Given a set M, a distance d = d M is a functionsuch that1. d(x, x) = 0,2. if d(x, y) = 0 then x = y,3. d(x, y) = d(y, x) (d is symmetric)d : M × M → [0, ∞]4. d(x, z) ≤ d(x, y) + d(y, z) (the triangular inequality).There are some possible generalizations.• The second request may be waived, in this case d is a semidistance.• The third request may be waived: then d would be an asymmetricdistance. Most theorems we will see can be generalized to asymmetricdistances; see [34].20