Appendix A. Basic Mathematical Concepts A.1 Hilbert and Banach Spaces (a) Hilbert Spaces Let X be a real vector space. An inner product on X is a function X×X → R, denoted by (u, v), that satisfies the following properties: (i) (u, v)=(v, u) for all u, v ∈ X; (ii) (αu + βv,w)=α(u, w)+β(v, w) for all α, β ∈ R and all u, v, w ∈ X; (iii) (u, u) ≥ 0 for all u ∈ X; (iv) (u, u) = 0 implies u =0. Two elements u, v ∈ X are said to be orthogonal in X if (u, v) = 0. The inner product (u, v) defines a norm on X by the relation u =(u, u)1/2 for all u ∈ X. The distance between two elements u, v ∈ X is the positive number u − v . A Cauchy sequence in X is a sequence {uk | k =0, 1,...} of elements of X that satisfies the following property: for each positive number ε>0, there exists an integer N = N(ε) > 0 such that the distance uk − um between any two elements of the sequence is smaller than ε provided both k and m are larger than N(ε). A sequence in X is said to converge to an element u ∈ X if the distance uk − u tends to 0 as k tends to ∞. A Hilbert space is a vector space equipped with an inner product for which all the Cauchy sequences are convergent. Examples n n (i) R endowed with the Euclidean product (u, v)= uivi is a finite- i=1 dimensional Hilbert space. 490 Appendix A. Basic Mathematical Concepts (ii) If [a, b] ⊂ R is an interval, the space L2(a, b) (see (A.6)) is an infinite- dimensional Hilbert space for the inner product b (u, v)= u(x)v(x)dx. a If X is a complex vector space, the inner product on X will be a complex- valued function. Then condition (i) has to be replaced by (i) (u, v)=(v, u) for all u, v ∈ X. (b) Banach Spaces The concept of Banach space extends that of Hilbert space. Given a vector space X,anorm on X is a function X → R, denoted by u , that satisfies the following properties: u + v ≤ u + v for all u, v ∈ X ; λu = |λ| u for all u ∈ X, and all λ ∈ R ; u ≥0 for all u ∈ X ; u =0 ifandonlyifu =0. A Banach space is a vector space equipped with a norm for which all the Cauchy sequences are convergent. Examples $ % n 1/p n p (i) R endowed with the norm u = |ui| (with 1 ≤ p<+∞)is i=1 a finite-dimensional Banach space. (ii) If [a, b] ⊂ R is an interval and 1 ≤ p<+∞,thespaceLp(a, b)(see again (A.6)) is an infinite-dimensional Banach space for the norm $ %1/p b u = |u(x)|pdx . a (c) Dual Spaces Let X be a Hilbert or a Banach space. A linear form F : X → R is said to be continuous if there exists a constant C>0 such that |F (u)|≤C u for all u ∈ X. A.2 The Cauchy-Schwarz Inequality 491 The set of all the linear continuous forms on X is a vector space. We can define a norm on this space by setting F (u) F X =sup . u∈X u u=0 The vector space of all the linear continuous forms on X is called the dual space of X and is denoted by X. Endowed with the previous norm, it is itself a Banach space. The bilinear form from X × X into R defined by F, u = F (u) is called the duality pairing between X and X. (d) The Riesz Representation Theorem If X is a Hilbert space, the dual space X can be canonically identified with X (hence, it is a Hilbert space). In fact, the Riesz representation theorem states that for each linear continuous form F on X, there exists a unique element u ∈ X such that F, v =(u, v) for all v ∈ X. Moreover, F X = u X. The Lax-Milgram Theorem (A.3) extends this result to the case in which (u, v) is replaced by a non-symmetric bilinear form a(u, v). A.2 The Cauchy-Schwarz Inequality Let X be a Hilbert space, endowed with the inner product (u, v)andthe associated norm u (see (A.1.a)). The Cauchy-Schwarz inequality states that |(u, v)|≤ u v for all u, v ∈ X. Of particular importance in the analysis of numerical methods for partial differential equations is the Cauchy-Schwarz inequality in the Lebesgue space L2(Ω), where Ω is a domain in Rn (see (A.9.h)). The previous inequality becomes: 1/2 1/2 2 2 u(x)v(x)dx ≤ u (x)dx v (x)dx Ω Ω Ω for all functions u, v ∈ L2(Ω). 492 Appendix A. Basic Mathematical Concepts A.3 The Lax-Milgram Theorem Let V be a real Hilbert space (see (A.1.a)). Let a : V × V → R be a bilinear continuous form on V , i.e., a satisfies (i) a(λu + µv, w)=λa(u, w)+µa(v, w)and a(u, λv + µw)=λa(u, v)+µa(u, w) for all u, v, w ∈ V and all λ, µ ∈ R; (ii) there exists a constant β>0 such that |a(u, v)|≤β u V v V for all u, v ∈ V . Assume that the form a is V -coercive,orV -elliptic, i.e., (iii) there exists a constant α>0 such that ≥ 2 ∈ a(u, u) α u V for all u V. Then for each form F ∈ V (the dual space of V , see (A.1.c)), there exists a unique solution u ∈ V to the variational problem a(u, v)=F (v) for all v ∈ V. Moreover, the following inequality holds: 1 u ≤ F . V α V A.4 Dense Subspace of a Normed Space Let X be a Hilbert or a Banach space with norm v .LetS ⊂ X be a sub- space of X. S is said to be dense in X if for each element v ∈ X there exists a sequence {vn | n =0, 1,...} of elements vn ∈ S, such that v − vn −→0asn −→ ∞ . Thus, each element of X can be approximated arbitrarily well by elements of S, in the distance induced by the norm of X. For example, the subspace C0([a, b]) of the continuous functions on a bounded, closed interval [a, b] of the real line, is dense in L2(a, b), the space of the measurable square-integrable functions on (a, b). Indeed, for each function v ∈ L2(a, b)andeachn>0, one can find a continuous function 0 vn ∈ C ([a, b]) such that b | − |2 ≤ 1 v(x) vn(x) dx 2 . a n A.6 The Spaces Lp(Ω), 1 ≤ p ≤ +∞ 493 A.5 The Spaces Cm(Ω), m ≥ 0 Let Ω be an open subset of Rd, with sufficiently smooth boundary. Let us denote by Ω the closure of Ω. For each multi-index α =(α1,...,αd)of | | ··· α |α| α1 αd nonnegative integers, set α = α1 + + αd and D v = ∂ v/∂x1 ...∂xd . We denote by Cm(Ω) the vector space of the functions v : Ω → R such that for each multi-index α with 0 ≤|α|≤m, Dαv exists and is continuous on Ω. Since a continuous function on a closed, bounded set is bounded there, one can set | α | v Cm(Ω) =supsup D v(x) . 0≤|α|≤m x∈Ω This is a norm for which Cm(Ω) is a Banach space (see (A.1.b)). The space C∞(Ω) is the space of the infinitely differentiable functions on Ω. Thus, a function v belongs to C∞(Ω) if and only if it belongs to Cm(Ω) for all m>0. A.6 The Spaces Lp(Ω), 1 ≤ p ≤ +∞ Let Ω denote a bounded, open domain in Rd,ford ≥ 1. For p<+∞,wedenoteby Lp(Ω) the space of the measurable functions → R | |p ∞ u : Ω such that Ω u(x) dx < + . It is a Banach space for the norm 1/p p u Lp(Ω) = |u(x)| dx . Ω Let L∞(Ω) be the Banach space of the measurable functions u : Ω → R that are bounded outside a set of measure zero, equipped with the norm u L∞(Ω) = ess sup |u(x)| . x∈Ω The space L2(Ω) is a Hilbert space for the inner product (u, v)= u(x)v(x)dx , Ω which induces the norm 1/2 2 u L2(Ω) = |u(x)| dx . Ω One can define spaces Lp(Ω) of complex functions in a straight-forward man- ner. 494 Appendix A. Basic Mathematical Concepts A.7 Infinitely Differentiable Functions and Distributions d Let Ω be a bounded, open domain in R ,ford =1, 2or3.Ifα =(α1,...,αd) is a multi-index of nonnegative integers, let us set ∂α1+···+αd v Dαv = . α1 ··· αd ∂x1 ∂xd We denote by D(Ω) the vector space of all the infinitely differentiable func- tions φ : Ω → R, for which there exists a closed set K ⊂ Ω such that φ ≡ 0 outside K. We say that a sequence of functions φn ∈ D(Ω) converges in D(Ω)to a function φ ∈ D(Ω)asn →∞, if there exists a common closed set K ⊂ Ω α α such that all the φn vanish outside K,andD φn → D φ uniformly on K as n →∞, for all multi-indices α.