(5) Appendix A Linear operators in Banach Spaces The goal of this appendix is to recall fundamental results on linear operators in Banach and Hilbert spaces; see, e.g., Aubin [15], Brezis [56], Lax [166], Rudin [206], Yosida [251], Zeidler [252] for further reading. The results collected in this appendix provide a theoretical framework for the mathematical analysis of the finite element method. A.1 Banach and Hilbert spaces To stay general, we consider complex vector spaces, i.e., vector spaces over the field C of complex numbers. The case of real vector spaces is recovered by replacing the field C by R, by removing the real part symbol ( ) and the complex conjugate symbol , and by interpreting the symbol asℜ the· absolute value instead of the modulus.· Some additional minor adaptations|·| (regarding duality and inner products) are indicated whenever relevant. A.1.1 Banach spaces Let V be a complex vector space. Definition A.1 (Norm). A norm on V is a map V : V [0, ) satisfy- ing the following three properties: k·k → ∞ (i) Definiteness: v =0 v =0. k kV ⇐⇒ (ii) 1-homogeneity: λv V = λ v V , for all λ C 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 ∈ A seminorm on V is a map : V [0, ) satisfying only (ii) and (iii). |·|V → ∞ Definition A.2 (Banach space). Equip V with a norm V ; then V is called a Banach space if every Cauchy sequence (with respectk·k to the metric d(x, y)= x y ) in V has a limit in V . k − kV 424 Appendix A. Linear operators in Banach Spaces Definition A.3 (Equivalent norms). Two norms V,1 and V,2 are said to be equivalent on V if there exists a positive real numberk·k c suchk·k that 1 c v v c− v , v V. (A.1) k kV,2 ≤k kV,1 ≤ k kV,2 ∀ ∈ One readily verifies that when (A.1) holds, V is a Banach space for the norm iff it is a Banach space for the norm . k·kV,1 k·kV,2 Remark A.4 (Finite dimension). If the vector space V is finite-dimensional, all the norms in V are equivalent. This result is false in infinite-dimensional vector spaces. Actually, the unit ball in V is a compact set (for the norm topology) if and only if V is finite-dimensional; see Brezis [56, Thm. 6.5], Lax [166, 5.2]. § ⊓⊔ A.1.2 Bounded linear maps Let V, W be complex vector spaces. A map A : V W is said to be linear → if A(v1 + v2) = A(v1)+ A(v2) for all v1, v2 V and A(λv) = λA(v) for all λ C and all v V . To alleviate the notation,∈ we simply write Av instead ∈ ∈ of A(v). Whenever V and W can be equipped with norms V and W , respectively, the linear map A is said to be bounded or continuousk·k if k·k Av W A (V ;W ) := sup k k < . (A.2) k kL v V v V ∞ ∈ k k In this book, we systematically abuse the notation by implicitly assuming that the argument in this type of supremum or infimum is nonzero. Proposition A.5 ( (V ; W )). The complex vector space composed of the L bounded linear maps from V to W is denoted (V ; W ). The map (V ;W ) defined by (A.2) is indeed a norm on (V ; W ),L and if W is a Banachk·kL space, then (V ; W ) equipped with this normL is also a Banach space. L Proof. See Rudin [206, p. 87], Yosida [251, p. 111]. ⊓⊔ Example A.6 (Continuous embedding). Assume that V W and that ⊂ there is a real number c such that v W c v V for all v V . Then the embedding of V into W is continuous.k Wek say≤ thatk kV is continuously∈ embedded . into W and we write V ֒ W → ⊓⊔ Whenever V and W are Banach spaces, maps in (V ; W ) are often called operators. The following result (often called UniformL Boundedness Principle) is a useful tool to study the limit of a sequence of operators. Theorem A.7 (Banach–Steinhaus). Let V, W be Banach spaces and let Ai i I be a family (not necessarily countable) in (V ; W ). Assume that { } ∈ L supi I Aiv W is a finite number for all v V . Then, there is a real number C such∈ k thatk ∈ sup Aiv W C v V , v V. (A.3) i I k k ≤ k k ∀ ∈ ∈ Appendices 425 Proof. See Brezis [56, p. 32], Lax [166, Chap. 10]. ⊓⊔ Corollary A.8 (Pointwise convergence). Let V, W be Banach spaces. Let (An)n N be a sequence in (V ; W ) such that, for all v V , the sequence ∈ L ∈ (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.7. Owing to (A.3), 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.9 (Uniform convergence on compact sets). Corollary A.8 does not claim that (An)n N converges to A in (V ; W ), i.e., uniformly on ∈ L bounded sets. However, a standard argument shows 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 ); C is finite owing to Corollary A.8(i). The ∈ k kL set K being compact, we infer that there is a finite set of points xi i I in K { 1} ∈ such that, for all v K, there is i I such that v x (3C)− ǫ. Owing ∈ ∈ k − ikV ≤ to the pointwise convergence of (An)n N to A, there is Ni such that, for all 1 ∈ 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. ∈ ≥ ∈ ⊓⊔ A.1.3 Compact operators Definition A.10 (Compact operator). Let V, W be Banach spaces. The operator T (V ; W ) is said to be compact if from every bounded sequence ∈ L (vn)n N in V one can extract a subsequence (vn )k N such that the sequence ∈ k ∈ (T vnk )k N converges in W ; equivalently, T maps the unit ball in V into a relatively∈ compact set in W . Theorem A.11 (Schauder). A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact. Example A.12 (Compact embedding). Assume that V W and that the embedding 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 . ∈ ⊓⊔ Proposition A.13 (Composition). Let W , X, Y , Z be four Banach spaces and let A (Z; Y ), K (Y ; X), B (X; W ). Assume that K is compact; then the operator∈ L B K∈ LA is compact.∈ L ◦ ◦ 426 Appendix A. Linear operators in Banach Spaces A.1.4 Duality Let V, W be complex vector spaces. The map A : V W is said to be → antilinear if A(v1 + v2)= Av1 + Av2 for all v1, v2 V and A(λv) = λAv for all λ C and all v V . The reason for considering∈ antilinear maps is that the weak∈ form of complex-valued∈ PDEs is derived by multiplying the PDE by the complex conjugate of a test function; therefore, the right-hand side of the weak formulation is an antilinear form. Alternatively, one could work with linear maps only upon noticing that if A is an antilinear map, then A (defined by Av := Av for all v V ) is a linear map. ∈ Definition A.14 (Dual space). The dual space of V is denoted V ′ and is composed of the bounded antilinear forms from V to C. An element A V ′ is called bounded antilinear form, and its action on an element v V is denoted∈ ∈ either Av or A, v ′ . h iV ,V Owing to Proposition A.5, V ′ is a Banach space with the norm Av A, v V ′,V A V ′ = sup | | = sup |h i |, A V ′. (A.4) k k v V v V v V v V ∀ ∈ ∈ k k ∈ k k In the real case, the absolute value can be omitted from the numerators since v can be considered. In the complex case, the modulus can be replaced by the± real part since v can be multiplied by any unit complex number. Theorem A.15 (Hahn–Banach). Let V be a normed vector space over C and let W be a subspace of V .