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

• If the second request is waived, then ‖ · ‖ is a seminorm.• If the third request holds only for λ ≥ 0, then the norm is asymmetric;in this case, it may happen that ‖x‖ ≠ ‖ − x‖.Each (semi/asymmetric)norm ‖ · ‖ defines a (semi/asymmetric)distanceSo a norm induces a topology.d(x, y) := ‖x − y‖.3.2.1 Examples of spaces of functionsWe present some examples and definitions.Definition 3.7 A locally-convex topological vector space E (shortenedas l.c.t.v.s. in the following) is a vector space equipped with a collection ofseminorms ‖ · ‖ k (with k ∈ K an index set); the seminorms induce a topology,such that c n → c iff ‖c n − c‖ k → n 0 for all k; and all vector space operationsare continuous w.r.to this topology.The simplest example of l.c.t.v.s. is obtained when there is only one norm; thisgives raise to two renowned examples of spaces.Definition 3.8 (Banach and Hilbert spaces) • A Banach space is avector space E with a norm ‖ · ‖ defining a distance d(x, y) := ‖x − y‖ suchthat E is metrically complete.• A Hilbert space is a space with an inner product 〈f, g〉, that defines anorm ‖f‖ := √ 〈f, g〉 such that E is metrically complete.(Note that a Hilbert space is also a Banach space).Example 3.9 Let I ⊂ lR k be open; let p ∈ [1, ∞]. A standard example ofBanach space is the L p space of functions f : I → lR n . If p ∈ [1, ∞), the L pspace contains all functions such that |f| p is Lebesgue integrable, and is equippedwith the norm√ ∫‖f‖ L p := p |f(x)| p dx .IFor the case p = ∞, L ∞ contains all Lebesgue measurable functions f : I →lR n such that there is a constant c ≥ 0 for which |f(x)| ≤ c for all x ∈ I \ N,where N is a set of measure zero; and L ∞ is equipped with the norm‖f‖ L ∞ := supess I |f(x)|that is the lowest possible constant c.If p = 2, L 2 is a Hilbert space by inner product∫〈f, g〉 := f(x) · g(x) dx .INote that, in these spaces, by definition, f = g iff the set {f ≠ g} hasLebesgue measure zero.22