Part XII, Chapter A Banach and Hilbert Spaces The goal of this appendix is to recall fundamental results on Banach and Hilbert spaces. The results collected herein provide a theoretical framework for the mathematical analysis of the finite element method. Some classical results are stated without proof; see Aubin [24], Brezis [97], Lax [321], Rudin [399], Yosida [483], Zeidler [486] for further insight. One important outcome of this appendix is the characterization of bijective operators in Banach spaces. To get started, let us recall the following definition of injective, surjective, and bijective maps. Definition A.1 (Injection, surjection, bijection). Let E and F be two sets. A function (or map) f : E F is said to be injective if every element of the codomain (i.e., F ) is mapped−→ to by at most one element of the domain (i.e., E). The function is said to be surjective if every element of the codomain is mapped to by at least one element of the domain. Finally, f is bijective if every element of the codomain is mapped to by exactly one element of the domain (i.e., f is both injective and surjective). A.1 Normed vector spaces Definition A.2 (Norm). Let V be a vector space over the field K = R or C. A norm on V is a map : V v v [0, ), (A.1) k·kV ∋ 7−→ k kV ∈ ∞ satisfying the following three properties: (i) Definiteness: v = 0 v = 0. k kV ⇐⇒ (ii) 1-homogeneity: λv V = λ v V , for all λ K and all v V . (iii) Triangle inequality:k k v + |w| k k v + w∈ , for all v,w∈ V . k kV ≤ k kV k kV ∈ 774 Appendix A. Banach and Hilbert Spaces A seminorm on V is a map from V to [0, ) which satisfies only properties (ii) and (iii). ∞ Definition A.3 (Equivalent norms). Two norms V,1 and V,2 are said to be equivalent on V if there exists a positive numberk·kc such thatk·k 1 c v v c− v , v V. (A.2) k kV,2 ≤ k kV,1 ≤ k kV,2 ∀ ∈ Remark A.4 (Finite dimension). If the vector space V has finite di- mension, all the norms in V are equivalent. This result is false in infinite- dimensional vector spaces. ⊓⊔ Proposition A.5 (Compactness of unit ball). Let V be a normed vector space and let B(0, 1) be the closed unit ball in V . Then, B(0, 1) is compact (for the norm topology) if and only if V is finite-dimensional. Proof. See Brezis [97, Thm. 6.5], Lax [321, 5.2]. § ⊓⊔ Definition A.6 (Bounded linear maps). Let V and W be two normed vector spaces. (V ; W ) is the vector space of bounded linear maps from V to W . The actionL of A (V ; W ) on an element v V is denoted A(v) or, more simply, Av. Maps∈ inL (V ; W ) are often called∈operators. L Example A.7 (Continuous embedding). Let V and W be two normed vector spaces. Assume that V W and that there is c such that v ⊂ k kW ≤ c v V for all v V . This property means that the embedding of V into W isk continuous.k We∈ say that V is continuously embedded into W and we write . V ֒ W → A.2 Banach spaces Definition A.8 (Banach space). A vector space V equipped with a norm V such that every Cauchy sequence (with respect to the metric d(x,y) = k·kx y ) in V has a limit in V is called a Banach space. k − kV A.2.1 Operators in Banach spaces Proposition A.9 (Banach space). Let V be a normed vector space and let W be a Banach space. Equip (V ; W ) with the norm L A(v) W A (V ;W ) = sup k k , A (V ; W ). (A.3) k kL v V v V ∀ ∈L ∈ k k Then, (V ; W ) is a Banach space. L Proof. See Rudin [399, p. 87], Yosida [483, p. 111]. ⊓⊔ Part XII. Appendices 775 Remark A.10 (Notation). In this book, we systematically abuse the no- A(v) W A(v) W tation by writing supv V k v k instead of supv V 0 k v k . ∈ k kV ∈ \{ } k kV ⊓⊔ The Uniform Boundedness Principle (or Banach–Steinhaus Theorem) is a useful tool to study the limit of a sequence of operators in Banach spaces. Theorem A.11 (Uniform Boundedness Principle). Let V and W be two Banach spaces. Let Ai i I be a family (not necessarily countable) of operators in (V ; W ). Assume{ that} ∈ L sup Aiv W < , v V. (A.4) i I k k ∞ ∀ ∈ ∈ Then, there is a constant C such that A v C v , v V, i I. (A.5) k i kW ≤ k kV ∀ ∈ ∀ ∈ Proof. See Brezis [97, p. 32], Lax [321, Chap. 10]. ⊓⊔ Corollary A.12 (Point-wise convergence). Let V and W be two Banach spaces. Let (An)n N be a sequence of operators in (V ; W ) such that, for all ∈ L v V , the sequence (Anv)n N converges as n to a limit in W denoted ∈ ∈ → ∞ Av (this means that the sequence (An)n N converges pointwise to A). Then, the following holds: ∈ (i) supn N An (V ;W ) < ; (ii) A ∈ (Vk; Wk)L; ∞ ∈L (iii) A (V ;W ) lim infn An (V ;W ). k kL ≤ →∞ k kL Proof. Statement (i) is just a consequence of Theorem A.11. Owing to (A.5), we infer that A v C v for all v V and all n N. Letting n , k n kW ≤ k kV ∈ ∈ → ∞ we obtain that Av W C v V , and since A is obviously linear, we infer that statement (ii)k holds.k ≤ Finally,k k statement (iii) results from the fact that Anv W An (V ;W ) v V for all v V and all n N. k k ≤ k kL k k ∈ ∈ ⊓⊔ Remark A.13 (Uniform convergence on compact sets). Note that Corollary A.12 does not claim that (An)n N converges to A in (V ; W ), i.e., uniformly on bounded sets. However, a∈ standard argument showsL that (An)n N converges uniformly to A on compact sets. Indeed, let K V be a ∈ ⊂ compact set. Let ǫ > 0. Set C := supn N An (V ;W ); this quantity is finite owing to statement (i) in Corollary A.12.∈ k K kbeingL compact, we infer that there is a finite set of points xi i I in K such that, for all v K, there is { } ∈1 ∈ i I such that v xi V (6C)− ǫ. Owing to the pointwise convergence of ∈ k − k ≤ 1 (An)n N to A, there is Ni such that, for all n Ni, Anxi Axi W 3 ǫ. Using∈ the triangle inequality and statement (iii)≥ above,k we infer− thatk ≤ A v Av A (v x ) + A x Ax + A(v x ) ǫ, k n − kW ≤ k n − i kW k n i − ikW k − i kW ≤ for all v K and all n maxi I Ni. ∈ ≥ ∈ ⊓⊔ 776 Appendix A. Banach and Hilbert Spaces Compact operators are encountered in various important situations, e.g., the Peetre–Tartar Lemma A.53 and the spectral theory developed in A.5.1. § Definition A.14 (Compact operator). Let V and W be two Banach spaces. T (V ; W ) is called a compact operator if from every bounded ∈ L sequence (vn)n N in V , one can extract a subsequence (vn )k N such that the ∈ k ∈ sequence (Tvnk )k N converges in W ; equivalently, T maps the unit ball in V into a relatively compact∈ set in W . Proposition A.15 (Composition with compact operator). Let W , X, Y , Z be four Banach spaces and A (Z; Y ), K (Y ; X), B (X; W ). Assume that K is compact. Then B∈ LK A is compact.∈ L ∈ L ◦ ◦ Example A.16 (Compact injection). A classical example is the case where V and W are two Banach spaces such that the injection of V into W is compact. Then from every bounded sequence (vn)n N in V , one can extract a subsequence that converges in W . ∈ ⊓⊔ A.2.2 Duality We start with real vector spaces and then discuss the extension to complex vector spaces. Definition A.17 (Dual space, Bounded linear forms). Let V be a normed vector space over R. The dual space of V is defined to be (V ; R) L and is denoted V ′. An element A V ′ is called a bounded linear form. Its action on an element v V is either∈ denoted A(v) (or Av) or by means of duality brackets in the form∈ A, v for all v V . h iV ′,V ∈ Owing to Proposition A.9, V ′ is a Banach space when equipped with the norm A(v) A, v V ′,V A V ′ = sup | | = sup |h i |, A V ′. (A.6) k k v V v V v V v V ∀ ∈ ∈ k k ∈ k k Note that the absolute value can be omitted from the numerators since A is linear and R-valued, and v can be considered in the supremum. ± Theorem A.18 (Hahn–Banach). Let V be a normed vector space over R and let W be a subspace of V . Let B W ′ = (W ; R) be a bounded linear B(w)∈ L map with norm B W ′ = supw W w . Then, there exists a bounded linear k k ∈ k kV form A V ′ with the following properties: ∈ (i) A is an extension of B, i.e., A(w)= B(w) for all w W .