A.1 Normed vector spaces

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1 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 v V [0, ), (A.1) satisfying the following three properties: (i) Definiteness: v V = 0 v = 0. (ii) 1-homogeneity: λv V = λ v V, for all λ K and all v V. (iii) Triangle inequality: v +w V v V + w V, for all v,w V.

2 774 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 number c such that c v V,2 v V,1 c 1 v V,2, v V. (A.2) Remark A.4 (Finite dimension). If the vector space V has finite dimension, all the norms in V are equivalent. This result is false in infinitedimensional 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. L(V;W) is the vector space of bounded linear maps from V to W. The action of A L(V;W) on an element v V is denoted A(v) or, more simply, Av. Maps in L(V;W) are often called operators. 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 W c v V for all v V. This property means that the embedding of V into W is continuous. 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) = x y V ) in V has a limit in V is called a Banach space. 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 L(V;W) with the norm A L(V;W) = sup v V Then, L(V;W) is a Banach space. A(v) W v V, A L(V;W). (A.3) Proof. See Rudin [399, p. 87], Yosida [483, p. 111].

3 Part XII. Appendices 775 Remark A.10 (Notation). In this book, we systematically abuse the notation by writing sup W A(v) A(v) v V v V instead of sup W v V\{0} v V. 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 {A i } i I be a family (not necessarily countable) of operators in L(V;W). Assume that Then, there is a constant C such that sup A i v W <, v V. (A.4) i I A i v W C v V, v V, i I. (A.5) 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 (A n ) n N be a sequence of operators in L(V;W) such that, for all v V, the sequence (A n v) n N converges as n to a limit in W denoted Av (this means that the sequence (A n ) n N converges pointwise to A). Then, the following holds: (i) sup n N A n L(V;W) < ; (ii) A L(V;W); (iii) A L(V;W) liminf n A n L(V;W). Proof. Statement (i) is just a consequence of Theorem A.11. Owing to (A.5), we infer that A n v W C v V for all v V and all n N. Letting n, we obtain that Av W C v V, and since A is obviously linear, we infer that statement (ii) holds. Finally, statement (iii) results from the fact that A n v W A n L(V;W) v V for all v V and all n N. Remark A.13 (Uniform convergence on compact sets). Note that Corollary A.12 does not claim that (A n ) n N converges to A in L(V;W), i.e., uniformly on bounded sets. However, a standard argument shows that (A n ) n N converges uniformly to A on compact sets. Indeed, let K V be a compact set. Let ǫ > 0. Set C := sup n N A n L(V;W) ; this quantity is finite owing to statement (i) in Corollary A.12. K being compact, we infer that there is a finite set of points {x i } i I in K such that, for all v K, there is i I such that v x i V (6C) 1 ǫ. Owing to the pointwise convergence of (A n ) n N to A, there is N i such that, for all n N i, A n x i Ax i W 1 3 ǫ. Using the triangle inequality and statement (iii) above, we infer that A n v Av W A n (v x i ) W + A n x i Ax i W + A(v x i ) W ǫ, for all v K and all n max i I N i.

4 776 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 L(V;W) is called a compact operator if from every bounded sequence (v n ) n N in V, one can extract a subsequence (v nk ) k N such that the sequence (Tv nk ) 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 L(Z;Y), K L(Y;X), B L(X;W). Assume that K is compact. Then B K A is compact. 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 (v n ) 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 L(V;R) 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 V,V for all v 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, A V. (A.6) v V v V v V v V 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 = L(W;R) be a bounded linear B(w) map with norm B W = sup w W w V. Then, there exists a bounded linear form A V with the following properties: (i) A is an extension of B, i.e., A(w) = B(w) for all w W. (ii) A V = B W. Proof. See Brezis [97, p. 3], Lax [321, Chap. 3], Rudin [399, p. 56], Yosida [483, p. 102]. The above statement is a simplified version of the actual Hahn Banach Theorem.

5 Part XII. Appendices 777 Corollary A.19 (Dual characterization of norm). Let V be a normed vector space over R. Then, the following holds: v V = sup A(v) = sup A,v V V, A V, A V =1 A V, A V =1 (A.7) for all v V, and the supremum is attained. Proof. Assume v 0 (the assertion is obvious for v = 0). We first observe that sup A V, A V =1A(v) v V. Let W = span(v) and let B W be defined as B(tv) = t v V for all t R. Owing to the Hahn Banach Theorem, there exists A V such that A V = B W = 1 and A(v) = B(v) = v V. Corollary A.20 (Characterization of density). Let V be a normed space over R and let W be a subspace of V. Assume that any bounded linear form in V vanishing identically on W vanishes identically on V. Then, W = V. Proof. See Brezis [97, p. 8], Rudin [399, Thm. 5.19]. Definition A.21 (Adjoint operator). Let V and W be two normed vector spaces over R and let A L(V;W). The adjoint operator, or dual operator, A : W V is defined by A w,v V,V = w,av W,W, (v,w ) V W. (A.8) Definition A.22 (Double dual). The double dual of a Banach space V over R is the dual of V and is denoted V. Proposition A.23 (Isometric embedding into V ). Let V be a Banach space over R. Then, V is a Banach space, and the linear map J V : V V defined by is an isometry. J V v,w V,V = w,v V,V, (v,w ) V V, (A.9) Proof. That V is a Banach space results from Proposition A.9. That J V is an isometry results from J V v V = sup J V v,w V,V = sup w,v V,V = v V, w V w V w V =1 w V =1 where the last equality results from Corollary A.19. Remark A.24 (Map J V ). Since the map J V is an isometry, it is injective. As a result, V can be identified with the subspace J V (V) V. It may happen that the map J V is not surjective. In this case, the space V is a proper subspace of V. For instance, L (D) = L 1 (D) but L 1 (D) L (D) with strict inclusion; see B.4 or Brezis [97, 4.3].

6 778 Appendix A. Banach and Hilbert Spaces Definition A.25 (Reflexive Banach spaces). Let V be a Banach space over R. V is said to be reflexive if J V is an isomorphism. Let now V be a normed vector space over C. The notion of dual space of V can be defined as in Definition A.17 by setting V = L(V;C). However, in the context of weak formulations of PDEs with complex-valued functions, it is more convenient to work with maps A : V C that are antilinear; this means that A(v +w) = Av +Aw for all v,w V (as usual), but A(λv) = λv for all λ C and all v V, where λ denotes the complex conjugate of λ (instead of A(λv) = λv, in which case the map is linear). We denote by V the vector space of antilinear maps that are bounded with respect to the norm (A.6) (note that we are now using the modulus in the numerators). Our aim is to extend the result of Corollary A.19 to measure the norm of the elements of V by the action of the elements of V. To this purpose, it is useful to consider V also as a vector space over R by restricting the scaling λv to λ R and v V. The corresponding vector space is denoted V R to distinguish it from V (thus, V and V R are the same sets, but equipped with different structures). For instance, if V = C m so that dim(v) = m, then dim(v R ) = 2m; a basis of V is the set {e k } 1 k m with e k,l = δ kl (the Kronecker symbol) for all l {1:m}, while a basis of V R is the set {e k,ie k } 1 k m with i 2 = 1. Another example is V = L 2 (0,2π;C) for which an Hilbertian basis is the set {cos(nx),sin((n+1)x)} n N, while an Hilbertian basis of V R is the set {cos(nx),icos(nx),sin((n+1)x),isin((n+1)x)} n N. Let V R be the dual space of V R, i.e., spanned by bounded R-linear maps from V to R. Lemma A.26 (Isometry for V ). The map I : V A I(A) V R such that I(A)(v) = R(A(v)) for all v V, is a bijective isometry. Proof. The operator I(A) maps onto R and is linear since I(A)(tv) = R(A(tv)) = R(tA(v)) = tr(a(v)) = ti(a)(v) for all t R and all v V. Moreover, I(A) is bounded since I(A)(v) = R(A(v)) A(v) A V v V, for all v V, so that I(A) V R A V. Furthermore, the map I is injective because R(A(v)) = 0 for all v V implies R(A(iv)) = 0, i.e., I(A(v)) = 0 so that A(v) = 0. Let us now prove that I is surjective. Let ψ V R and consider the map A : V C so that A(v) = ψ(v)+iψ(iv), v V. (Recall that ψ is only R-linear.) By construction, I(A) = ψ, and the map A : V C is antilinear; indeed, for all λ C, writing λ = µ + iν with µ,ν R, we infer that

7 Part XII. Appendices 779 A(λw) = ψ(µw+iνw)+iψ(iµw νw) = µψ(w)+νψ(iw)+iµψ(iw) iνψ(w) = µ(ψ(w)+iψ(iw)) iν(ψ(w)+iψ(iw)) = λa(v), for all w V, where we have used the R-linearity of ψ. Let us finally show that A V ψ V R. Let v V be such that A(v) 0 and set λ = A(v) A(v) C. Then, A(v) = λ 1 A(v) = A(λ 1 v) = ψ(λ 1 v)+iψ(iλ 1 v), but since ψ takes values in R, we infer that ψ(iλ 1 v) = 0, so that A(v) = ψ(λ 1 v). As a result, A(v) ψ V R λ 1 v V = ψ V R v V, since λ = 1. This concludes the proof. Corollary A.27 (Dual characterization of norm). Let V be a normed vector space over C. Then, the following holds: v V = max R(A(v)). A V, A V =1 (A.10) for all v V. Proof. Combine the result of Corollary A.19 with Lemma A.26. Remark A.28 (Use of modulus). Note that it is possible to replace (A.10) by v V = max A V, A V =1 A(v) since it is always possible to multiply A in the supremum by a unitary complex number so that A(v) is real and nonnegative. Remark A.29 (Hahn Banach). A version of the Hahn Banach Theorem A.18 in complex vector spaces can be derived similarly to the above construction; see Lax [321, p. 27]. The rest of the material is adapted straightforwardly. The adjoint of an operator A L(V;W) is still defined by (A.8), and one can verify that it maps (linearly) bounded antilinear maps in W to bounded antilinear maps in V. Moreover, the bidual is defined by considering bounded antilinear forms on V, and the linear isometry extending that from Proposition A.23 is such that J V v,w V,V = w,v V,V. A.2.3 Interpolation between Banach spaces Interpolating between Banach spaces is a useful tool to bridge between known results so as to derive new results that could difficult to obtain directly. An important application is the derivation of interpolation error estimates in fractional-order Sobolev spaces. There are many interpolation methods; see,

8 780 Appendix A. Banach and Hilbert Spaces e.g., Bergh and Löfström [47], Tartar [443] and references therein. For simplicity, we focus here on the real interpolation K-method; see [47, 3.1] and [443, Chap. 22]. Let V 0 and V 1 be two normed vector spaces, continuously embedded into a common topological vector space V. Then, V 0 +V 1 is a normed vector space with the (canonical) norm v V0+V 1 = inf v=v0+v 1 ( v 0 V0 + v 1 V1 ). Moreover, if V 0 and V 1 are Banach spaces, then V 0 +V 1 is also a Banach space; see [47, Lem ]. For all v V 0 +V 1 and all t > 0, define K(t,v) = inf v=v 0+v 1 ( v 0 V0 +t v 1 V1 ). (A.11) For all t > 0, v K(t,v) defines a norm on V 0 +V 1 equivalent to the canonical norm. One can also verify that t K(t,v) is nondecreasing and concave (and therefore continuous) and that t 1 tk(t,v) is increasing. Definition A.30 (Interpolated space). Let θ (0,1) and let p [1, ]. The interpolated space [V 0,V 1 ] θ,p is defined to be [V 0,V 1 ] θ,p = {v V 0 +V 1 t θ K(t,v) Lp (R +; dt t ) < }, (A.12) where ϕ L p (R +; dt t ) = ( ) ϕ(t) p dt 1 p 0 t for p [1, ) and ϕ L (R +; dt t ) = sup 0<t< ϕ(t). This space is equipped with the norm v [V0,V 1] θ,p = t θ K(t,v) Lp (R +; dt t ). If V 0 and V 1 are Banach spaces, so is [V 0,V 1 ] θ,p. Remark A.31 (Value for θ). Since K(t,v) min(1,t) v V0+V 1, the space [V 0,V 1 ] θ,p reducesto{0}ift θ min(1,t) L p (R + ; dt t ).Inparticular,[V 0,V 1 ] θ,p is trivial if θ {0,1} and p <. Remark A.32 (Gagliardo set). Themapt K(t,v)hasasimplegeometric interpretation. Introducing the Gagliardo set G(v) = {(x 0,x 1 ) R 2 v = v 0 +v 1 with v 0 V0 x 0 and v 1 V1 x 1 }, one can verify that G(v) is convex and that K(t,v) = inf v G(v) (x 0 +tx 1 ), so that the map t K(t,v) is one way to explore the boundary of G(v); see [47, p. 39]. Remark A.33 (Intersection). The vector space V 0 V 1 can be equipped with the (canonical) norm v V0 V 1 = max( v V0, v V1 ). For all v V 0 V 1, one can verify that K(t,v) min(1,t) v V0 V 1, whence we infer the continuous embedding V 0 V 1 [V 0,V 1 ] θ,p for all θ (0,1) and p [1, ]. As a result, if V 0 V 1, then V 0 [V 0,V 1 ] θ,p. Lemma A.34 (Continuous embedding). Let θ (0,1) and p,q [1, ] with p q. Then, [V 0,V 1 ] θ,p [V 0,V 1 ] θ,q.

9 Part XII. Appendices 781 Theorem A.35 (Riesz Thorin, interpolation of operators). Let A : V 0 + V 1 W 0 + W 1 be a linear operator that maps V 0 and V 1 boundedly to W 0 and W 1. Then, for all θ (0,1) and all p [1, ], A maps [V 0,V 1 ] θ,p boundedly to [W 0,W 1 ] θ,p. Moreover, A L([V0,V 1] θ,p ;[W 0,W 1] θ,p ) A 1 θ L(V 0;W 0) A θ L(V 1;W 1). Proof. See [443, Lem. 22.3]. (A.13) Theorem A.36 (Lions Peetre, reiteration). Let θ 0,θ 1 [0,1] with θ 0 θ 1. Assume that [V 0,V 1 ] θ0,1 W 0 [V 0,V 1 ] θ0, and [V 0,V 1 ] θ1,1 W 1 [V 0,V 1 ] θ1,. Then, for all θ (0,1) and all p [1, ], [W 0,W 1 ] θ,p = [V 0,V 1 ] η,p with equivalent norms where η = (1 θ)θ 0 +θθ 1. Proof. See Tartar [443, Thm. 26.2]. (JLG) I changed θ 0,θ 1 (0,1) to θ 0,θ 1 [0,1] since this is what we need in general. Theorem A.37 (Lions Peetre, extension). Let V 0, V 1, F be three Banach spaces. Let A L(V 0 V 1 ;F), then A extends into a linear continuous mapping from [V 0,V 1 ] θ,1;j to F if and only if c <, v V 0 V 1 Proof. See [443, Lem. 25.3]. Av F c v 1 θ V 0 v θ V 1. (A.14) Theorem A.38 (Interpolation of dual spaces). Let θ (0, 1) and p [1, ). Then, [V 0,V 1 ] θ,p = [V 1,V 0] 1 θ,p where p = p p 1 (with the convention that p = if p = 1). Proof. See Bergh and Löfström [47].page! A.3 Hilbert spaces We start with real vector spaces and then briefly discuss the extension to complex vector spaces. Definition A.39 (Inner product). Let V be a vector space over R. An inner product (or scalar product) on V is a map (, ) V : V V (v,w) (v,w) V R, (A.15) satisfying the following three properties: (i) Bilinearity: (v,w) V is a linear function of w V for fixed v V, and it is a linear function of v V for fixed w V. (ii) Symmetry: (v,w) V = (w,v) V for all v,w V. (iii) Positive definiteness: (v,v) V 0 for all v V and (v,v) V = 0 v = 0.

10 782 Appendix A. Banach and Hilbert Spaces Proposition A.40 (Cauchy Schwarz). Let (, ) V be an inner product on the real vector space V. By setting v V = (v,v) 1 2 V, v V, (A.16) one defines a norm on V. Moreover, the Cauchy Schwarz 1 inequality holds: (v,w) V v V w V, v,w V. (A.17) Remark A.41 (Equality). The Cauchy Schwarz inequality can be seen as a v V w V consequence of the identity v V w V (v,w) V = v 2 v V w 2 w, V V valid for all non-zero v,w in V. This identity shows that equality holds in (A.17) if and only if v and w are collinear. Proposition A.42 (Arithmetic-geometric inequality). Let x 1,...,x n be non-negative numbers. Then, (x 1 x 2...x n ) 1 n 1 n (x x n ). Proof. Use the convexity of the function x e x. (A.18) This inequality is frequently used in conjunction with the Cauchy Schwarz inequality. In particular, it implies that (v,w) V γ 2 v 2 V + 1 2γ w 2 V, γ > 0, v,w V. (A.19) Definition A.43 (Hilbert spaces). A Hilbert space V is an inner product space over R that is complete with respect to the induced norm (and is, therefore, a Banach space). The inner product is denoted (, ) V and the induced norm V. A Hilbert space is said to be separable if it admits a countable and dense subset. Theorem A.44 (Riesz Fréchet). Let V be a Hilbert space over R. For each v V, there exists a unique u V such that v,w V,V = (u,w) V, w V. (A.20) Moreover, the map v V u V is an isometric isomorphism. Proof. See Brezis [97, Thm. 5.5], Lax [321, p. 56], Yosida [483, p. 90]. An important consequence of the Riesz Fréchet Theorem is the following: Proposition A.45 (Reflexivity). Hilbert spaces are reflexive. Proof. Let V be a Hilbert space. The Riesz Fréchet Theorem implies that V can be identified with V ; similarly, V can be identified with V. 1 Augustin-Louis Cauchy ( ) and Herman Schwarz ( )

11 Part XII. Appendices 783 Proposition A.46 (Orthogonal projection). Let V be a Hilbert space over R. Let U be a non-empty, closed, and convex subset of V. Let f V. (i) There is a unique u U such that f u V = min v U f v V. (ii) This unique minimizer is characterized by the Euler Lagrange condition (f u,v u) V 0 for all v U. (iii) In the case where U is a subspace of V, the unique minimizer is characterized by the condition (f u,v) V = 0 for all v U, and the map Π U : V f u U is linear with Π U L(V;U) = 1 (unless U = {0} so that Π U L(V;U) = 0). Proof. See Exercise Let now V be a vector space over C. Then, an inner product on V is a map (, ) V : V V (v,w) (v,w) V C satisfying the following three properties: (i) Sesquilinearity:(v,w) V isanantilinearfunctionofw V forfixedv V, and it is a linear function of v V for fixed w V. (ii) Hermitian symmetry: (v,w) V = (w,v) V for all v,w V. (iii) Positive definiteness: (v,v) V 0 for all v V and (v,v) V = 0 v = 0 (note that (v,v) V is always real owing to Hermitian symmetry). The extension of the above results from real to complex Hilbert spaces is as follows. The map v (v,v) 1/2 V still defines a norm on V, and the Cauchy Schwarz inequality still takes the form (A.17) (with the modulus on the lefthand side). The Riesz Fréchet Theorem A.44 still states that, for all v V (recall that v is antilinear by convention), there exists a unique u V such that v,w V,V = (u,w) V for all w V, and the map v V u V is an isometric (linear) isomorphism. Finally, the Euler Lagrange condition from Proposition A.46 now becomes R(f u,v u) V 0 for all v U. The proof of these extensions hinges on the fact that, if V is a complex Hilbert space equipped with the inner product (, ) V, then V R equipped with the inner product R(, ) V is a real Hilbert space (recall that V and V R are the same sets, equipped with different structures). Let us prove for instance the Riesz Fréchet Theorem. Let v V. Using the bijective isometry from Lemma A.26, we consider I(v ) V R. Owing to Theorem A.44 on V R equipped with R(, ) V, there is a unique u V R such that R(u,w) V = I(v )(w) = R v,w V,V for all w V R. Considering w and iw yields (u,w) = v,w V,V for all w V. A.4 Bijective Banach operators Let V and W be two Banach spaces. Maps in L(V;W) are called (linear) Banach operators. This section presents classical results to characterize bijective linear Banach operators, see Aubin [24], Brezis [97], Yosida [483]. Some

12 784 Appendix A. Banach and Hilbert Spaces of the material presented herein is adapted from Azerad [26], Guermond and Quartapelle [264]. For simplicity, we implicitly assume that V and W are real Banach spaces, and we briefly indicate relevant changes in the complex case. A.4.1 Fundamental results For A L(V;W), we denote by ker(a) its kernel and by im(a) its range. The operator A being bounded, ker(a) is closed in V. Hence, the quotient of V by ker(a), V/ker(A), can be defined. This space is composed of equivalence classes ˇv suchthatv andw areinthesameclass ˇv ifandonlyifv w ker(a). Theorem A.47 (Quotient space). The space V/ker(A) is a Banach space when equipped with the norm ˇv = inf v ˇv v V. Moreover, defining Ǎ : V/ker(A) im(a) by Ǎˇv = Av for all v in ˇv, Ǎ is an isomorphism. Proof. See Brezis [97, 11.2], Yosida [483, p. 60]. For subspaces M V and N V, we introduce the so-called annihilators of M and N which are defined as follows: M = {v V m M, v,m V,V = 0}, N = {v V n N, n,v V,V = 0}. (A.21) (A.22) A characterization of ker(a) and im(a) is given by the following: Lemma A.48 (Kernel and range). For A in L(V;W), the following properties hold: (i) ker(a) = (im(a )). (ii) ker(a ) = (im(a)). (iii) im(a) = (ker(a )). (iv) im(a ) (ker(a)). Proof. See Brezis [97, Cor. 2.18], Yosida [483, p ]. Showing that the range of an operator is closed is a crucial step towards proving that this operator is surjective. This is the purpose of the following fundamental theorem: Theorem A.49 (Banach or Closed Range). Let A L(V;W). The following statements are equivalent: (i) im(a) is closed. (ii) im(a ) is closed. (iii) im(a) = (ker(a )). (iv) im(a ) = (ker(a)). Proof. See Brezis [97, Thm. 2.19], Yosida [483, p. 205].

13 Part XII. Appendices 785 We now put in place the second keystone of the edifice: Theorem A.50 (Open Mapping). If A L(V;W) is surjective and U is an open set in V, then A(U) is open in W. Proof. See Brezis [97, Thm. 2.6], Lax [321, p. 168], Rudin [399, p. 47], Yosida [483, p. 75]. Theorem A.50, also due to Banach, has far-reaching consequences. In particular, we deduce the following: Lemma A.51 (Characterization of closed range). Let A L(V; W). The following statements are equivalent: (i) im(a) is closed. (ii) There exists α > 0 such that w im(a), v w V, Av w = w and α v w V w W. (A.23) Proof. (i) (ii). Since im(a) is closed in W, im(a) is a Banach space. Applying the Open Mapping Theorem to A : V im(a) and U = B V (0,1) (the open unit ball in V) yields that A(B V (0,1)) is open in im(a). Since 0 A(B V (0,1)), there is γ > 0 such that B W (0,γ) A(B V (0,1)). Let w w W B W (0,α), there is z B V (0,1) such that w im(a). Since γ 2 Az = γ w 2 w W. Setting v = 2 w W γ z leads Av = w and γ 2 v V w W. (ii) (i). Let (w n ) n N be a sequence in im(a) that converges to some w W. Using (A.23), we infer that there exists a sequence (v n ) n N in V such that Av n = w n and α v n V w n W. Then, (v n ) n N is a Cauchy sequence in V. Since V is a Banach space, (v n ) n N converges to a certain v V. Owing to the boundedness of A, (Av n ) n N converges to Av. Hence, w = Av im(a), proving statement (i). Remark A.52 (Bounded inverse). A first consequence of Lemma A.51 is that if A L(V;W) is bijective, then its inverse is necessarily bounded. Indeed, the fact that A is bijective implies that A is injective and im(a) is closed. Lemma A.51 implies that there is α > 0 such that A 1 w V 1 α w W, i.e., A 1 is bounded. Let us finally give a sufficient condition for the image of an injective operator to be closed. Lemma A.53 (Peetre Tartar). Let X, Y, Z be three Banach spaces. Let A L(X;Y) be an injective operator and let T L(X;Z) be a compact operator. If there is c > 0 such that c x X Ax Y + Tx Z, then im(a) is closed; equivalently, there is α > 0 such that x X, α x X Ax Y. (A.24)

14 786 Appendix A. Banach and Hilbert Spaces Proof. By contradiction. Assume that there is a sequence (x n ) n N of X such that x n X = 1 and Ax n Y converges to zero when n goes to infinity. Since T is compact and the sequence (x n ) n N is bounded, there is a subsequence (x nk ) k N such that (Tx nk ) k N is a Cauchy sequence in Z. Owing to the inequality α x nk x mk X Ax nk Ax mk Y + Tx nk Tx mk Z, (x nk ) k N is a Cauchy sequence in X. Let x be its limit. Clearly, x X = 1. The boundedness of A implies Ax nk Ax and Ax = 0 since Ax nk 0. Since A is injective x = 0, which contradicts the fact that x X = 1. A.4.2 Characterization of surjectivity As a consequence of the Closed Range Theorem and of the Open Mapping Theorem, we deduce two lemmas characterizing surjective operators. Lemma A.54 (Surjectivity of A ). Let A L(V;W). The following statements are equivalent: (i) A : W V is surjective. (ii) A : V W is injective and im(a) is closed in W. (iii) There exists α > 0 such that v V, Av W α v V, (A.25) or, equivalently, there exists α > 0 such that inf sup w,av W,W v V w W w α. W v V In the complex case, real parts of duality brackets are considered. (A.26) Proof. (i) (iii). The Open Mapping Theorem implies that, for all v V, there is w v W such that A w v = v and w v W α 1 v V. Let now v V. Then, v,v V,V v V = A w v,v V,V v V α 1 w v,av W,W w v W α 1 w,av W,W sup w W w. W Taking the supremum in v V yields (A.26) since v,v V v V = sup,v v V v V α 1 w,av sup W,W w W w. W (iii) (ii). The bound (A.25) implies that A is injective. To prove that im(a) is closed, consider a sequence (v n ) n N such that (Av n ) n N is a Cauchy sequence in W. Then, (A.25) implies that (v n ) n N is a Cauchy sequence in V.

15 Part XII. Appendices 787 Let v be its limit. A being bounded implies that Av n Av; hence, im(a) is closed. (ii) (i). Since im(a) is closed, we use Theorem A.49(iv) together with the injectivity of A to infer that im(a ) = {0} = V. Lemma A.55 (Surjectivity of A). Let A L(V;W). The following statements are equivalent: (i) A : V W is surjective. (ii) A : W V is injective et im(a ) is closed in V. (iii) There exists α > 0 such that w W, A w V α w W, (A.27) or, equivalently, there exists α > 0 such that inf sup A w,v V,V w W v V w α. W v V (A.28) In the complex case, real parts of duality brackets are considered. Proof. Similar to that of Lemma A.54. Remark A.56 (Lions Theorem). The statement(i) (iii) in Lemma A.55 is sometimes referred to as Lions Theorem. Establishing the a priori estimate (A.28)isanecessaryandsufficientconditiontoprovethattheproblemAu = f has at least one solution u in V for all f in W. One easily verifies (see Lemma A.57) that (A.23) implies the inf-sup condition (A.28). In practice, however, it is often easier to check condition (A.28) than to prove that for all w im(a), there exists an inverse image v w satisfying (A.23). At this point, the natural question that arises is to determine whether the constant α in (A.28) is the same as that in (A.23). The answer to this question is the purpose of the next lemma which is due to Azerad [26, 27]. Lemma A.57 (Inf-sup condition). Let V and W be two Banach spaces and let A L(V;W) be a surjective operator. Let α > 0. The property w W, v w V, Av w = w and α v w V w W, (A.29) implies inf sup A w,v V,V w W v V w α. W v V (A.30) The converse is true if V is reflexive. In the complex case, it is the real part of the duality bracket that is considered.

16 788 Appendix A. Banach and Hilbert Spaces Proof. (1) The implication. By definition of the norm in W, w W, w W = sup w,w W,W. w W w W =1 For all w in W, there is v w V such that Av w = w and α v w V w W. Let w in W. Therefore, Hence, w,w W,W = w,av w W,W = A w,v w V,V 1 α A w V w W. w W = sup w,w W,W 1 α A w V. w W w W =1 The desired inequality follows from the definition of the norm in V. (2) Let us prove the converse statement under the assumption that V is reflexive. The inf-sup inequality being equivalent to A w V α w W for all w W, A is injective. Let v im(a ) and define z (v ) W such that A (z (v )) = v. Note that z (v ) is unique since A is injective; this in turn implies that z ( ) : im(a ) V W is a linear mapping. (Note also that z is injective: assume the 0 = z (v ), then 0 = A (z (v )) = v. The mapping is also surjective: let w W, then A (z (A w )) = A w, which implies z (A w ) = w since A is surjective.) In conclusion z ( ) : im(a ) V W is an isomorphism. Let w W and let us construct an inverse image for w, say v w, satisfying (A.23). We first define the linear form φ w : im(a ) R by v im(a ), φ w (v ) = z (v ),w W,W, i.e., φ w (A w ) = w,w W,W for all w W. Hence, φ w (v ) z (v ) W w W 1 α A z (v ) V w W 1 α v V w W. This means that φ w is bounded on im(a ) equipped with the norm of V. Owing to the Hahn Banach Theorem, φ w can be extended to V with the samenorm.let φ w V betheextensioninquestionwith φ w V 1 α w W. Since V is assumed to be reflexive, there is v w V such that J V (v w ) = φ w. As a result, w W, w,av w W,W = A w,v w V,V = J V (v w ),A w V,V = φ w,a w V,V = φ w(a w ) = z (A w ),w W,W = w,w W,W, showing that Av w = w. Hence, v w is an inverse image of w and v w V = J V (v w ) V = φ w V 1 α w W. Remark A.58 (Linearity). Note that the dependence of v w with respect to w in Lemma A.57 is a priori nonlinear. Linearity can be obtained when the setting is Hilbertian by using the zero extension of φ w on the orthogonal complement of im(a ) instead of invoking the Hahn Banach Theorem.

17 Part XII. Appendices 789 A.4.3 Characterization of bijectivity The following theorem provides the theoretical foundation of the BNB Theorem of Theorem A.59 (Bijectivity of A). Let A L(V;W). The following statements are equivalent: (i) A : V W is bijective. (ii) A is injective, im(a) is closed, and A : W V is injective. (iii) A is injective and there exists α > 0 such that or, equivalently, A is injective and Av W α v V, v V, (A.31) inf sup w,av W,W v V w W w =: α > 0. W v V (A.32) In the complex case, real parts of duality brackets are considered. Proof. (1) Statements (ii) and (iii) are equivalent since (A.31) is equivalent to A injective and im(a) closed owing to Lemma A.54. (2) Let us first prove that (i) implies (ii). Since A is surjective, ker(a ) = im(a) = {0}, i.e., A is injective. Since im(a) = W is closed and A is injective, this yields (ii). Finally, to prove that (ii) implies (i), we only need to prove that (ii) implies the surjectivity of A. The injectivity of A implies im(a) = (ker(a )) = W. Since im(a) is closed, im(a) = W, i.e., A is surjective. Remark A.60 (Bijectivity of A ). The bijectivity of A L(V;W) is equivalent to that of A L(W ;V ). Indeed, statement (ii) in Theorem A.59 is equivalent to A injective and A surjective owing to the equivalence of statements (i) and (ii) from Lemma A.54. Corollary A.61 (Inf-sup condition). Let A L(V;W) be a bijective operator. Assume that V is reflexive. Then, inf sup w,av W,W v V w W w = inf W v sup w,av W,W V w W v V w. W v V (A.33) In the complex case, real parts of duality brackets are considered. Proof. The left-hand side, l, and the right-hand side, r, of (A.33) are two positive finite numbers, since A is a bijective bounded operator. The lefthand side being equal to l means that l is the largest number such that Av W l v V for all v in V. Let w W and w W. Since A is surjective, there is v w V so that Av w = w and the previous statement regarding l implies that l v w V w W. This in turn implies that

18 790 Appendix A. Banach and Hilbert Spaces w w,w W,W W = sup w W w W A w V sup w W w,av w W,W = sup w W w W v w V w W 1 l A w V, A w,v w V,V = sup w W w W which implies l r. That r l is proved similarly by working with W in lieu of V, V in lieu of W and A in lieu of A. The above reasoning leads to inf w W v,a w V,V sup v V v V w W v,a w V,V inf sup v V w W v V w W, and we conclude using the reflexivity of V. Remark A.62 (Counter-example). Note that (A.33) may not hold if A 0 is not bijective. For instance if A : (x 1,x 2,x 3...) (0,x 1,x 2,x 3,...) is the right shift operator in l 2, then A : (x 1,x 2,x 3...) (x 2,x 3,x 4,...) is the left shift operator. It can be verified that A is injective but not surjective whereas A is injective but not surjective. It can also be shown that l = 1 and r = 0. A.4.4 Coercive operators We now focus on the smaller class of coercive operators. Definition A.63 (Coercive operator). Let V be a Banach space over R. A L(V;V ) is said to be a coercive operator if there exist a number α > 0 and ξ = ±1 such that ξ Av,v V,V α v 2 V, v V. (A.34) In the complex case, A L(V;V ) is said to be a coercive operator if there exist a real number α > 0 and a complex number ξ with ξ = 1 such that R(ξ Av,v V,V) α v 2 V, v V. (A.35) The following proposition shows that the notion of coercivity is relevant only in Hilbert spaces: Proposition A.64 (Hilbert structure). Let V be a Banach space. V can be equipped with a Hilbert structure with the same topology if and only if there is a coercive operator in L(V;V ). Proof. See Exercise Corollary A.65 (Sufficient condition). Coercivity is a sufficient condition for an operator A L(V;V ) to be bijective. Proof. Corollary A.65 is the Lax Milgram Lemma; see 17.2.

19 Part XII. Appendices 791 We now introduce the class of self-adjoint operators. Definition A.66 (Self-adjoint operator). Let V be a reflexive Banach space, so that V and V are identified. The operator A L(V;V ) is said to be self-adjoint if A = A in the real case and if A = A in the complex case. Self-adjoint bijective operators are characterized as follows: Corollary A.67 (Self-adjoint bijective operator). Let V be a reflexive Banach space and let A L(V;V ) be a self-adjoint operator. Then, A is bijective if and only if there is a number α > 0 such that Av V α v V, v V. (A.36) Proof. Owing to Theorem A.59, the bijectivity of A implies that A satisfies inequality (A.36). Conversely, inequality (A.36) means that A is injective. It follows that A is injective since A = A (or A = A) by hypothesis. The conclusion is then a consequence of Theorem A.59(iii). We finally introduce the concept of monotonicity. Definition A.68 (Monotone operator). Let V be a Banach space over R. The operator A L(V;V ) is said to be monotone if Av,v V,V 0, v V. In the complex case, the condition becomes R( Av,v V,V) 0 for all v V. Corollary A.69 (Equivalent condition). Let V be a reflexive Banach space and let A L(V;V ) be a monotone self-adjoint operator. Then, A is bijective if and only if A is coercive (with ξ = 1). Proof. See Exercise A.5 Spectral theory The rest of this chapter still to be checked We briefly recall in this section some essential facts regarding the spectral theory of linear operators. The material is classical and can be found in Brezis [97, Chap. 6], Chatelin [131, p ], Dunford and Schwartz [201, Part I, pp ], Lax [321, Chap. 21&32]. Definition A.70 (Resolvent, spectrum). Let L be a complex Banach space and let T L(L;L). The resolvent set, ρ(t), and the spectrum of T, σ(t), are the sets in C such that

20 792 Appendix A. Banach and Hilbert Spaces ρ(t) = {z C; (zi T) 1 L(L;L)}, σ(t) = C\ρ(T). (A.37) (A.38) µ C is called eigenvalue of T if ker(µi T) {0} and ker(µi T) is the associated eigenspace. The set of eigenvalues is denoted ε(t). Theorem A.71 (Gelfand). Let T L(L;L), then (i) ρ(t) and σ(t) are nonempty. (ii) σ(t) is compact in C. (iii) σ(t) = lim n T n 1 n L(L;L) where σ(t) := max λ σ(t) λ is called the spectral radius of T. A.5.1 Compact operators We start with the following important results regarding compact operators. Theorem A.72 (Schauder). A bounded linear operator between Banach spaces is compact if and only if its adjoint is. Theorem A.73 (Fredholm alternative). Let T L(L; L) be a compact operator and µ C\{0}. µi T is injective if and only if µi T is surjective. The Fredholm alternative is often reformulated in the following equivalent way: Either µi T is bijective or ker(µi T) 0. The key result for compact operators is the following (see, e.g., [321, p. 238]). Theorem A.74 (Spectrum). Let T L(L;L) be a compact operator, then (i) σ(t) is a countable set with no accumulation point other than zero. (ii) Each nonzero member of σ(t) is an isolated eigenvalue. (iii) For each nonzero µ σ(t), there is a smallest integer α, called ascent, with the property that of ker(µi T) α = ker(µi T) α+1. Then dimker(µi T) α is called the algebraic multiplicity of µ and dimker(µi T) is called the geometric multiplicity. (iv) µ σ(t) if and only if µ σ(t ). The ascent, algebraic multiplicity, and geometric multiplicity of µ σ(t)\{0} and µ are equal, respectively. The vectors in ker(µi T) are the eigenvectors associated with µ and those in ker(µi T) α are called generalized eigenvectors. Both ker(µi T) α and ker(µi T) are invariant under T. Note that the ascent of µi T is one and the two multiplicities are equal if T is self-adjoint; in this case the eigenvalues are real (see Theorem A.76). Denoting by g the geometric multiplicity of µ, it can be shown that α+g 1 m αg. Corollary A.75. Assume that diml = and let T L(L;L) be a compact operator. Then (i) 0 σ(t).

21 Part XII. Appendices 793 (ii) σ(t)\{0} = ε(t)\{0}. (iii) One of the following situations holds: (1) σ(t) = {0}; (2) σ(t)\{0} is finite; (3) σ(t)\{0} is a sequence that converges to 0. A.5.2 Symmetric operators in Hilbert spaces Assume that L is a Hilbert space and let T L(L;L). The operator T is said to be symmetric if (Tv,w) H = (v,tw) H for all v,w H; equivalently, identifying L and L, T is self-adjoint, i.e., T = T. The key result for symmetric operators is the following (see, e.g., [321, p. 356]). Theorem A.76 (Real spectrum, spectral radius). Let L be a Hilbert space and let T L(L;L) be a symmetric operator. Then, σ(t) R and {a,b} σ(t) [a,b], (A.39) with a = inf v H, v H=1(Tv,v) H and b = sup v H, v H=1(Tv,v) H. Moreover, T L(L;L) = σ(t) = max( a, b ). As a consequence of Corollary A.67, we infer the following Corollary A.77 (Characterization of σ(t)). Let L be a Hilbert space and let T L(L;L) be a symmetric operator. Then λ σ(t) if and only if there is a sequence (v n ) n N in L such that v n L = 1 for all n N and Tv n λv n 0 as n. We conclude this section by considering symmetric compact operators. Proposition A.78 (Symmetric compact operator). Let L be a Hilbert space and let T L(L;L) be a symmetric compact operator. Then L has a Hilbertian basis composed of eigenvectors of T.