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

• The Fréchet space of smooth functions C ∞ (I → lR n ).The seminorms are‖f‖ k = sup |f (k) (x)|x∈Iwhere f (k) is the k-th derivative. In this space, f n → f iff all the derivativesf n(k) (x) converge to derivatives f (k) (x), and for any fixed k the convergenceis uniform w.r.to x ∈ I.This last is the strongest topology between the topologies usually associated tospaces of functions.3.2.4 Dual spaces.Definition 3.12 Given a l.c.t.v.s. E, the dual space E ∗ is the space of alllinear functions L : E → lR.If E is a Banach space, it is easy to see that E ∗ is again a Banach space, withnorm‖L‖ E ∗ := sup‖x‖ E ≤1|Lx| .The biggest problem when dealing with Fréchet spaces, is that the dual ofa Fréchet space is not in general a Fréchet space, since it often fails to bemetrizable. (In most cases, the duals of Fréchet spaces are “quite wide” spaces;a classical example being the dual elements of smooth functions, that are thedistributions). So given F, G Fréchet spaces, we cannot easily work with “thespace L(F, G) of linear functions between F and G”.As a workaround, given an auxiliary space H, we will consider “indexedfamilies of linear maps” L : F × H → G, where L(·, h) is linear, and L is jointlycontinuous; but we will not consider L as a maph ↦→ (f ↦→ L(f, h))H → L(F, G)(3.1)3.2.5 DerivativesAn example is the Gâteaux differential.Definition 3.13 We say that a continuous map P : U → G, where F, G areFréchet spaces and U ⊂ F is open, is Gâteaux differentiable if for any h ∈ Fthe limitP (f + th) − P (f)DP (f; h) := lim= [∂ t P (f + th)]|t→0 tt=0(3.2)exists. The map DP (f; ·) : F → G is the Gâteaux differential.Definition 3.14 We say that P is C 1 if DP : U × F → G exists and is jointlycontinuous.24