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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, ). Let E and F be two sets. A (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 (). Let V be a 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 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 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 (). 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 (). 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 We start with real vector spaces and then discuss the extension to complex vector spaces.

Definition A.17 (, 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 . 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 . (ii) A = B . ∈ k kV ′ k kW ′ 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. ⊓⊔ 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.7) k k A V , A =1 A V , A =1h i ∈ ′ k kV ′ ∈ ′ k kV ′ for all v V , and the supremum is attained. ∈ Proof. Assume v = 0 (the assertion is obvious for v = 0). We first observe that 6 sup A(v) v . Let W = span(v) and let B W ′ be defined A V ′, A V =1 V as B(∈tv)k =kt ′v for all≤ kt k R. Owing to the Hahn–Banach∈ Theorem, there k kV ∈ exists A V ′ such that A = B = 1 and A(v)= B(v)= v . ∈ k kV ′ k kW ′ k kV ⊓⊔ 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 (V ; W ). The adjoint operator, or dual operator, ∈L A∗ : W ′ V ′ is defined by →

A∗w′,v = w′, Av , (v,w′) V W ′. (A.8) h iV ′,V h iW ′,W ∀ ∈ × 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 JV : V V ′′ defined by →

J v,w′ = w′,v , (v,w′) V V ′, (A.9) h V iV ′′,V ′ h iV ′,V ∀ ∈ × is an isometry.

Proof. That V ′′ is a Banach space results from Proposition A.9. That JV is an isometry results from

JV v V ′′ = sup JV v,w′ V ′′,V ′ = sup w′,v V ′,V = v V , k k w V |h i | w V |h i | k k ′∈ ′ ′∈ ′ w′ =1 w′ =1 k kV ′ k kV ′ where the last equality results from Corollary A.19. ⊓⊔

Remark A.24 (Map JV ). Since the map JV is an isometry, it is injective. As a result, V can be identified with the subspace J (V ) V ′′. It may V ⊂ happen that the map JV is not surjective. In this case, the space V is a 1 1 proper subspace of V ′′. For instance, L∞(D)= L (D)′ but L (D) ( L∞(D)′ with strict inclusion; see B.4 or Brezis [97, 4.3]. § § ⊓⊔ 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 JV is an .

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 ′ = (V ; C). However, in the context of weak formulations of PDEs with complex-valueL d 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 VR to ∈ ∈ distinguish it from V (thus, V and VR are the same sets, but equipped with different structures). For instance, if V = Cm so that dim(V ) = m, then dim(VR) = 2m; a of V is the set ek 1 k m with ek,l = δkl (the Kro- { } ≤ ≤ necker symbol) for all l 1:m , while a basis of VR is the set ek,iek 1 k m with i2 = 1. Another∈ example { } is V = L2(0, 2π; C) for which{ an Hilbertian} ≤ ≤ − basis is the set cos(nx), sin((n + 1)x) n N, while an Hilbertian basis of VR is { } ∈ the set cos(nx),i cos(nx), sin((n + 1)x),i sin((n + 1)x) n N. { } ∈ Let VR′ be the dual space of VR, 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) VR′ such that I(A)(v)= (A(v)) for all v V , is a bijective isometry.∋ 7→ ∈ ℜ ∈ Proof. The operator I(A) maps onto R and is linear since I(A)(tv) = (A(tv)) = (tA(v)) = t (A(v)) = tI(A)(v) for all t R and all v V . ℜMoreover, I(ℜA) is boundedℜ since ∈ ∈

I(A)(v)= (A(v)) A(v) A v , ℜ ≤| | ≤ k kV ′ k kV for all v V , so that I(A) A . Furthermore, the map I is injective VR′ V ′ because ∈(A(v)) = 0k for allkv ≤V kimpliesk (A(iv)) = 0, i.e., (A(v))=0 so ℜ ∈ ℜ ℑ that A(v) = 0. Let us now prove that I is surjective. Let ψ VR′ 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 ∈ ∈ 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 ∈ A(v) that A V ψ V . Let v V be such that A(v) = 0 and set λ = C. k k ′ ≤ k k R′ ∈ 6 A(v) ∈ Then, | | 1 1 1 1 A(v) = λ− A(v)= A(λ− v)= ψ(λ− v)+ iψ(iλ− v), | | 1 but since ψ takes values in R, we infer that ψ(iλ− v) = 0, so that A(v) = 1 | | ψ(λ− v). As a result,

1 A(v) ψ V λ− v V = ψ V v V , | | ≤ k k R′ k k k k R′ k k 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 (A(v)). (A.10) k k A V , A =1 ℜ ∈ ′ k kV ′ 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 = maxA V , A =1 A(v) since it is always possible to multiply A ′ V ′ in thek k supremum∈ by ak unitaryk | complex| number so that A(v) is real and non- negative. ⊓⊔ Remark A.29 (Hahn–Banach). A version of the Hahn–Banach Theo- rem 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 (V ; W ) is still defined by (A.8), and one can verify that it ∈ L 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,w′ = w ,v . h V iV ′′,V ′ h ′ iV ′,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, 780 Appendix A. Banach and Hilbert Spaces e.g., Bergh and L¨ofstr¨om [47], Tartar [443] and references therein. For sim- plicity, we focus here on the real interpolation K-method; see [47, 3.1] and [443, Chap. 22]. § Let V0 and V1 be two normed vector spaces, continuously embedded into a common . Then, V0 + V1 is a normed vector space with the (canonical) norm v V= inf ( v + v ). Moreover, k kV0+V1 v=v0+v1 k 0kV0 k 1kV1 if V0 and V1 are Banach spaces, then V0 + V1 is also a Banach space; see [47, Lem. 2.3.1]. For all v V + V and all t> 0, define ∈ 0 1

K(t,v)= inf ( v0 V0 + t v1 V1 ). (A.11) v=v0+v1 k k k k

For all t> 0, v K(t,v) defines a norm on V0 +V1 equivalent to the canonical norm. One can7→ also verify that t K(t,v) is nondecreasing and concave (and therefore continuous) and that t7→ 1 K(t,v) is increasing. 7→ t Definition A.30 (Interpolated space). Let θ (0, 1) and let p [1, ]. ∈ ∈ ∞ The interpolated space [V0,V1]θ,p is defined to be

θ [V0,V1]θ,p = v V0 + V1 t− K(t,v) Lp(R ; dt ) < , (A.12) { ∈ | k k + t ∞}

1 p dt p where ϕ Lp(R ; dt ) = ∞ ϕ(t) for p [1, ) and ϕ L (R ; dt ) = k k + t 0 | | t ∈ ∞ k k ∞ + t sup0

If V0 and V1 are Banach spaces, so is [V0,V1]θ,p.

Remark A.31 (Value for θ). Since K(t,v) min(1,t) v V0+V1 , the space θ p ≥ dt k k [V0,V1]θ,p reduces to 0 if t− min(1,t) L (R+; t ). In particular, [V0,V1]θ,p is trivial if θ 0, 1 {and} p< . 6∈ ∈ { } ∞ ⊓⊔ Remark A.32 (Gagliardo set). The map t K(t,v) has a simple geomet- 7→ 2 ric interpretation. Introducing the Gagliardo set G(v)= (x0,x1) R v = v +v with v x and v x , one can verify{ that G(v∈) is convex| 0 1 k 0kV0 ≤ 0 k 1kV1 ≤ 1} and that K(t,v) = infv ∂G(v)(x0 + tx1), so that the map t K(t,v) is one way to explore the boundary∈ of G(v); see [47, p. 39]. 7→ ⊓⊔ Remark A.33 (Intersection). The vector space V V can be equipped 0 ∩ 1 with the (canonical) norm v V V = max( v V , v V ). For all v V0 V1, k k 0∩ 1 k k 0 k k 1 ∈ ∩ one can verify that K(t,v) min(1,t) v V V , whence we infer the contin- ≤ k k 0∩ 1 uous embedding V0 V1 ֒ [V0,V1]θ,p for all θ (0, 1) and p [1, ]. As a ∞ ∋ ∋ . [ result, if V V , then∩ V →֒ [V ,V 0 ⊂ 1 0 → 0 1 θ,p ⊓⊔ Lemma A.34 (Continuous embedding). Let θ (0, 1) and p,q [1, ] ∞ ∋ ∋ . [ with p q. Then, [V ,V ] ֒ [V ,V ≤ 0 1 θ,p → 0 1 θ,q Part XII. Appendices 781

Theorem A.35 (Riesz–Thorin, interpolation of operators). Let A : V0 + V1 W0 + W1 be a linear operator that maps V0 and V1 boundedly to W and →W . Then, for all θ (0, 1) and all p [1, ], A maps [V ,V ] 0 1 ∈ ∈ ∞ 0 1 θ,p boundedly to [W0,W1]θ,p. Moreover,

1 θ θ A A − A . (A.13) ([V0,V1]θ,p;[W0,W1]θ,p) (V ;W ) (V1;W1) k kL ≤ k kL 0 0 k kL Proof. See [443, Lem. 22.3]. ⊓⊔ Theorem A.36 (Lions–Peetre, reiteration). Let θ0,θ1 [0, 1] with ∈ (JLG) I changed ֒ θ0 = θ1. Assume that [V0,V1]θ0,1 ֒ W0 ֒ [V0,V1]θ0, and [V0,V1]θ1,1 6 → → ∞ → θ0,θ1 (0, 1) to ∋ = W1 ֒ [V0,V1]θ1, . Then, for all θ (0, 1) and all p [1, ], [W0,W1]θ,p → ∞ ∈ ∈ ∞ θ0,θ1 [0, 1] since [V0,V1]η,p with equivalent norms where η = (1 θ)θ0 + θθ1. − this is∈ what we need Proof. See Tartar [443, Thm. 26.2]. in general. ⊓⊔ Theorem A.37 (Lions–Peetre, extension). Let V0, V1, F be three Banach spaces. Let A (V V ; F ), then A extends into a linear continuous mapping ∈L 0∩ 1 from [V0,V1]θ,1;J to F if and only if

1 θ θ c< , v V0 V1 Av F c v − v . (A.14) ∃ ∞ ∀ ∈ ∩ k k ≤ k kV0 k kV1 Proof. See [443, Lem. 25.3]. ⊓⊔ Theorem A.38 (Interpolation of dual spaces). Let θ (0, 1) and p p ∈ ∈ [1, ). Then, [V0,V1]θ,p′ =[V1′,V0′]1 θ,p′ where p′ = p 1 (with the convention ∞ − − that p′ = if p = 1). ∞ Proof. See Bergh and L¨ofstr¨om [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 product) on V is a map

( , ) : V V (v,w) (v,w) R, (A.15) · · V × ∋ 7−→ V ∈ satisfying the following three properties:

(i) Bilinearity: (v,w)V is a of w V for fixed v V , and it is a linear function of v V for fixed w V .∈ ∈ (ii) Symmetry: (v,w) =(w,v∈ ) for all v,w∈ V . V V ∈ (iii) Positive definiteness: (v,v)V 0 for all v V and (v,v)V = 0 v = 0. ≥ ∈ ⇐⇒ 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 · ·

1 v =(v,v) 2 , v V, (A.16) k kV V ∀ ∈ one defines a norm on V . Moreover, the Cauchy–Schwarz1 inequality holds:

(v,w) v w , v,w V. (A.17) | V | ≤ k kV k kV ∀ ∈ Remark A.41 (Equality). The Cauchy–Schwarz inequality can be seen as a v V w V v w 2 consequence of the identity v V w V (v,w)V = k k k k , k k k k − 2 v V − w V V valid for all non-zero v,w in V . This identity shows that equalityk k k holdsk in

(A.17) if and only if v and w are collinear. ⊓⊔

Proposition A.42 (Arithmetic-geometric inequality). Let x1,...,xn be non-negative numbers. Then,

1 1 (x x ...x ) n (x + ... + x ). (A.18) 1 2 n ≤ n 1 n Proof. Use the convexity of the function x ex. 7→ ⊓⊔ This inequality is frequently used in conjunction with the Cauchy–Schwarz inequality. In particular, it implies that

(v,w) γ v 2 + 1 w 2 , γ > 0, v,w V. (A.19) | V |≤ 2 k kV 2γ k kV ∀ ∀ ∈ Definition A.43 (Hilbert spaces). A V is an over R that is complete with respect to the induced norm (and is, there- fore, a Banach space). The inner product is denoted ( , ) and the induced · · V norm V . A Hilbert space is said to be separable if it admits a countable and densek·k subset.

Theorem A.44 (Riesz–Fr´echet). Let V be a Hilbert space over R. For each v′ V ′, there exists a unique u V such that ∈ ∈

v′,w =(u,w) , w V. (A.20) h iV ′,V V ∀ ∈

Moreover, the map v′ V ′ u V is an isometric isomorphism. ∈ 7→ ∈ Proof. See Brezis [97, Thm. 5.5], Lax [321, p. 56], Yosida [483, p. 90]. ⊓⊔ An important consequence of the Riesz–Fr´echet Theorem is the following: Proposition A.45 (Reflexivity). Hilbert spaces are reflexive.

Proof. Let V be a Hilbert space. The Riesz–Fr´echet Theorem implies that V can be identified with V ′; similarly, V ′ can be identified with V ′′. ⊓⊔ 1 Augustin-Louis Cauchy (1789–1857) and Herman Schwarz (1843–1921) 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 = minv U f v V . (ii) This unique minimizer∈ is characterizedk − byk the Euler–Lagran∈ k −gek 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 char- acterized by the condition (f u,v) = 0 for all v U, and the map − V ∈ ΠU : V f u U is linear with ΠU (V ;U) = 1 (unless U = 0 so ∋ 7→ ∈ k kL { } that ΠU (V ;U) = 0). k kL Proof. See Exercise 17.3. ⊓⊔ 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:· · × ∋ 7−→ ∈

(i) Sesquilinearity: (v,w)V is an antilinear function of w V for fixed v 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) 0 for all v V and (v,v∈ ) = 0 v = V ≥ ∈ 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 1/2 follows. The map v (v,v)V still defines a norm on V , and the Cauchy– Schwarz inequality still7→ takes the form (A.17) (with the modulus on the left- hand side). The Riesz–Fr´echet 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 anh isometrici (linear) isomorphism.∈ Finally, the Euler–Lagrange∈ 7→ condition∈ from Proposition A.46 now becomes (f u,v u)V 0 for all v U. The proof of these extensions hinges on theℜ fact− that,− if V≤is a complex∈ Hilbert space equipped with the inner product ( , ) , then VR equipped with the inner · · V product ( , )V is a real Hilbert space (recall that V and VR are the same sets, equippedℜ · · with different structures). Let us prove for instance the Riesz– Fr´echet Theorem. Let v′ V ′. Using the bijective isometry from Lemma A.26, ∈ we consider I(v′) V ′ . Owing to Theorem A.44 on VR equipped with ( , ) , ∈ R ℜ · · V there is a unique u VR such that (u,w) = I(v′)(w)= v′,w for all ∈ ℜ V ℜh iV ′,V w VR. Considering w and iw yields (u,w)= v′,w for all w V . ∈ h iV ′,V ∈

A.4 Bijective Banach operators

Let V and W be two Banach spaces. Maps in (V ; W ) are called (linear) Banach operators. This section presents classical resultsL to characterize bijec- tive linear Banach operators, see Aubin [24], Brezis [97], Yosida [483]. Some 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 (V ; W ), we denote by ker(A) its and by im(A) its range. The operator∈LA 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 classesv ˇ such that v and w are in the same classv ˇ if and only if v w ker(A). − ∈ Theorem A.47 (Quotient space). The space V/ker(A) is a Banach space when equipped with the norm vˇ = infv vˇ v V . Moreover, defining Aˇ : V/ker(A) im(A) by Aˇvˇ = Avkfork all v in∈vˇ,kAˇkis 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 = 0 , (A.21) { ∈ |∀ ∈ h iV ′,V } N ⊥ = v V n′ N, n′,v = 0 . (A.22) { ∈ |∀ ∈ h iV ′,V } A characterization of ker(A) and im(A) is given by the following: Lemma A.48 (Kernel and range). For A in (V ; W ), the following prop- erties hold: L

(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. 202-209]. ⊓⊔ 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 (V ; W ). The fol- lowing statements are equivalent: ∈ L (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]. ⊓⊔ Part XII. Appendices 785

We now put in place the second keystone of the edifice: Theorem A.50 (Open Mapping). If A (V ; W ) is surjective and U is an open set in V , then A(U) is open in W .∈ L

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 par- ticular, we deduce the following: Lemma A.51 (Characterization of closed range). Let A (V ; W ). The following statements are equivalent: ∈ L (i) im(A) is closed. (ii) There exists α> 0 such that

w im(A), v V, Av = w and α v w . (A.23) ∀ ∈ ∃ w ∈ w k wkV ≤ k kW Proof. (i) (ii). Since im(A) is closed in W , im(A) is a Banach space. Ap- plying the⇒ Open Mapping Theorem to A : V im(A) and U = B (0, 1) → V (the open unit ball in V ) yields that A(BV (0, 1)) is open in im(A). Since 0 A(BV (0, 1)), there is γ > 0 such that BW (0,γ) A(BV (0, 1)). Let ∈ γ w ⊂ w im(A). Since 2 w BW (0,α), there is z BV (0, 1) such that ∈ k kW ∈ ∈ γ w 2 w W γ Az = 2 w . Setting v = k γk z leads Av = w and 2 v V w W . k kW k k ≤ k k (ii) (i). Let (wn)n N be a sequence in im(A) that converges to some w W . ⇒ ∈ ∈ Using (A.23), we infer that there exists a sequence (vn)n N in V such that ∈ Avn = wn and α vn V wn W . Then, (vn)n N is a Cauchy sequence in V . k k ≤ k k ∈ Since V is a Banach space, (vn)n N converges to a certain v V . Owing to ∈ ∈ the boundedness of A, (Avn)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 (V ; W ) is bijective, then its inverse is necessarily bounded. Indeed, the∈ fact L that A is bijective implies that A is injective and im(A) 1 is closed. Lemma A.51 implies that there is α > 0 such that A− w V 1 1 k k ≤ w , i.e., A− is bounded. α k kW ⊓⊔ Let us finally give a sufficient condition for the of an injective op- erator to be closed.

Lemma A.53 (Peetre–Tartar). Let X, Y , Z be three Banach spaces. Let A (X; Y ) be an injective operator and let T (X; Z) be a compact ∈ L ∈ L 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 suchk k that≤ k k k k

x X, α x Ax . (A.24) ∀ ∈ k kX ≤ k kY 786 Appendix A. Banach and Hilbert Spaces

Proof. By contradiction. Assume that there is a sequence (xn)n N of X such that x = 1 and Ax converges to zero when n goes to infinity.∈ Since k nkX k nkY T is compact and the sequence (xn)n N is bounded, there is a subsequence ∈ (xnk )k N such that (Txnk )k N is a Cauchy sequence in Z. Owing to the in- equality∈ ∈ α x x Ax Ax + Tx Tx , k nk − mk kX ≤ k nk − mk kY k nk − mk kZ (xn )k N is a Cauchy sequence in X. Let x be its limit. Clearly, x X = 1. k ∈ k k The boundedness of A implies Axnk Ax and Ax = 0 since Axnk 0. Since A is injective x = 0, which contradicts→ the fact that x = 1. → k kX ⊓⊔ 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 (V ; W ). The following state- ments are equivalent: ∈L

(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 α v , (A.25) ∀ ∈ k kW ≥ k kV or, equivalently, there exists α> 0 such that

w′, Av inf sup h iW ′,W α. (A.26) v V w W w′ W v V ≥ ∈ ′∈ ′ k k ′ k k In the complex case, real parts of duality brackets are considered.

Proof. (i) (iii). The Open Mapping Theorem implies that, for all v′ V ′, there is w ⇒ W such that A w = v and w α 1 v . Let∈ now v′ ′ ′ ∗ v′ ′ ′ v′ ′ W ′ − ′ V ′ v V . Then,∈ k k ≤ k k ∈

v′,v V ′,V A∗wv′ ,v V ′,V 1 wv′ , Av W ′,W h i = h ′ i α− h ′ i v′ V v′ V ≤ w′ W k k ′ k k ′ k v′ k ′ 1 w′, Av W ′,W α− sup h i . ≤ w W w′ W ′∈ ′ k k ′ Taking the supremum in v′ V ′ yields (A.26) since ∈

v′,v V ′,V 1 w′, Av W ′,W v V = sup h i α− sup h i . k k v V v′ V ≤ w W w′ W ′∈ ′ k k ′ ′∈ ′ k k ′ (iii) (ii). The bound (A.25) implies that A is injective. To prove that im(A) ⇒ is closed, consider a sequence (vn)n N such that (Avn)n N is a Cauchy se- ∈ ∈ quence in W . Then, (A.25) implies that (vn)n N is a Cauchy sequence in V . ∈ Part XII. Appendices 787

Let v be its limit. A being bounded implies that Avn 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 (V ; W ). The following state- ments are equivalent: ∈L (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′ α w′ , (A.27) ∀ ∈ k kV ′ ≥ k kW ′ or, equivalently, there exists α> 0 such that

A∗w′,v inf sup h iV ′,V α. (A.28) w′ W ′ v V w′ W v V ≥ ∈ ∈ k k ′ k k 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) is a necessary and sufficient condition to prove that the problem Au = 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 con- dition (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 vw satis- fying (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 (V ; W ) be a surjective operator. Let α> 0. The property ∈L w W, v V, Av = w and α v w , (A.29) ∀ ∈ ∃ w ∈ w k wkV ≤ k kW implies A∗w′,v inf sup h iV ′,V α. (A.30) w′ W ′ v V w′ W v V ≥ ∈ ∈ k k ′ k k The converse is true if V is reflexive. In the complex case, it is the real part of the duality bracket that is considered. 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 . ∀ ∈ k k w W h i w ∈ =1 k kW For all w in W , there is v V such that Av = w and α v w . w ∈ w k wkV ≤ k kW Let w′ in W ′. Therefore, 1 w′,w = w′, Av = A∗w′,v A∗w′ w . h iW ′,W h wiW ′,W h wiV ′,V ≤ α k kV ′ k kW Hence, 1 w′ W ′ = sup w′,w W ′,W α A∗w′ V ′ . k k w W h i ≤ k k w ∈ =1 k kW 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 re- flexive. The inf-sup inequality being equivalent to A∗w′ α w′ for k kV ′ ≥ k kW ′ 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 map- ping 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 , satisfying (A.23). We first define the linear form φ : im(A∗) R by w w → v′ im(A∗), φ (v′)= z′(v′),w , ∀ ∈ w h iW ′,W i.e., φ (A∗w′)= w′,w for all w′ W ′. Hence, w h iW ′,W ∈ 1 1 φ (v′) z′(v′) w A∗z′(v′) w v′ w . | w | ≤ k kW ′ k kW ≤ α k kV ′ k kW ≤ α k kV ′ k kW This means that φw is bounded on im(A∗) equipped with the norm of V ′. Owing to the Hahn–Banach Theorem, φw can be extended to V ′ with the 1 same norm. Let φ V ′′ be the extension in question with φ w . w ∈ k wkV ′′ ≤ α k kW Since V is assumed to be reflexive, there is vw V such that JV (vw) = φw. As a result, e ∈ e e w′ W ′, w′, Av = A∗w′,v = J (v ), A∗w′ ∀ ∈ h wiW ′,W h wiV ′,V h V w iV ′′,V ′ = φ , A∗w′ = φ (A∗w′) h w iV ′′,V ′ w = z′(A∗w′),w W ′,W = w′,w W ′,W , he i h i showing that Avw = w. Hence, vw is an inverse image of w and v = J (v ) = φ 1 w . k wkV k V w kV ′′ k wkV ′′ ≤ α k kW ⊓⊔ Remark A.58 (Linearity). Note that the dependence of v with respect e w 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 φw on the orthogonal complement of im(A∗) instead of invoking the Hahn–Banach Theorem. ⊓⊔ Part XII. Appendices 789

A.4.3 Characterization of bijectivity The following theorem provides the theoretical foundation of the BNB Theo- rem of 17.3. § Theorem A.59 (Bijectivity of A). Let A (V ; W ). The following state- ments are equivalent: ∈L (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

Av α v , v V, (A.31) k kW ≥ k kV ∀ ∈

or, equivalently, A∗ is injective and

w′, Av inf sup h iW ′,W =: α> 0. (A.32) v V w W w′ W v V ∈ ′∈ ′ k k ′ k k 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 (V ; W ) is ∈ L equivalent to that of A∗ (W ′; V ′). Indeed, statement (ii) in Theorem A.59 ∈L 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 (V ; W ) be a bijective op- erator. Assume that V is reflexive. Then, ∈ L

w′, Av w′, Av inf sup h iW ′,W = inf sup h iW ′,W . (A.33) v V w W w′ W v V w′ W ′ v V w′ W v V ∈ ′∈ ′ k k ′ k k ∈ ∈ k k ′ k k 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 . The left- hand side being equal to l means that l is the largest number such that Av l v for all v in V . Let w′ W ′ and w W . Since A is surjective, k kW ≥ k kV ∈ ∈ there is vw V so that Avw = w and the previous statement regarding l implies that∈l v w . This in turn implies that k wkV ≤ k kW 790 Appendix A. Banach and Hilbert Spaces

w′,w W ′,W w′, Avw W ′,W A∗w′,vw V ′,V w′ W ′ = sup h i = sup h i = sup h i k k w W w W w W w W w W w W ∈ k k ∈ k k ∈ k k vw V 1 A∗w′ V ′ sup k k A∗w′ V ′ , ≤ k k w W w W ≤ l k k ∈ k k 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

v′′, A∗w′ v′′, A∗w′ inf sup h iV ′′,V ′ inf sup h iV ′′,V ′ , w′ W ′ v V v′′ V w′ W ≤ v′′ V ′′ w W v′′ V w′ W ∈ ′′∈ ′′ k k ′′ k k ′ ∈ ′∈ ′ k k ′′ k k ′ and we conclude using the reflexivity of V . ⊓⊔ Remark A.62 (Counter-example). Note that (A.33) may not hold if A = 6 0 is not bijective. For instance if A : (x1,x2,x3 ...) (0,x1,x2,x3,...) is 2 7−→ the right shift operator in ℓ , then A∗ : (x1,x2,x3 ...) (x2,x3,x4,...) is the left shift operator. It can be verified that A is injective7−→ 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 (V ; V ′) is said to be a coercive operator if there exist a number α > 0 and∈ξ L= 1 such that ± ξ Av,v α v 2 , v V. (A.34) h iV ′,V ≥ k kV ∀ ∈

In the complex case, A (V ; V ′) is said to be a coercive operator if there exist a α> ∈0 and L a ξ with ξ = 1 such that | | (ξ Av,v ) α v 2 , v V. (A.35) ℜ h iV ′,V ≥ k kV ∀ ∈ 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 (V ; V ′). L Proof. See Exercise 17.5. ⊓⊔ Corollary A.65 (Sufficient condition). Coercivity is a sufficient condition for an operator A (V ; V ′) to be bijective. ∈L Proof. Corollary A.65 is the Lax–Milgram Lemma; see 17.2. § ⊓⊔ 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 (V ; V ′) is said ∈ L 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 (V ; V ′) be a self-adjoint operator. Then, A is bijective if and only if there∈ Lis a number α> 0 such that

Av α v , v V. (A.36) k kV ′ ≥ k kV ∀ ∈ 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 (V ; V ′) is said to be monotone if ∈L Av,v 0, v V. h iV ′,V ≥ ∀ ∈ In the complex case, the condition becomes ( Av,v ) 0 for all v V . ℜ h iV ′,V ≥ ∈ Corollary A.69 (Equivalent condition). Let V be a reflexive Banach space and let A (V ; V ′) be a monotone self-adjoint operator. Then, A is bijective if and∈ only L if A is coercive (with ξ = 1).

Proof. See Exercise 17.6. ⊓⊔ The rest of this chapter still to be checked A.5 Spectral theory

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. 95-120], Dunford and Schwartz [201, Part I, pp. 577-580], Lax [321, Chap. 21&32].

Definition A.70 (Resolvent, spectrum). Let L be a complex Banach space and let T (L; L). The resolvent set, ρ(T ), and the spectrum of T , σ(T ), are the sets∈ in L C such that 792 Appendix A. Banach and Hilbert Spaces

1 ρ(T )= z C; (zI T )− (L; L) , (A.37) { ∈ − ∈L } σ(T )= C ρ(T ). (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− is6 denoted{ } ε(T ). −

Theorem A.71 (Gelfand). Let T (L; L), then ∈L (i) ρ(T ) and σ(T ) are nonempty. (ii) σ(T ) is compact in C. 1 n n (iii) σ(T ) = limn T where σ(T ) := maxλ σ(T ) λ is called the | | →∞ k k (L;L) | | ∈ | | spectral radius of T . L

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) be a compact operator and µ C 0 . µI T is injective if and only∈ Lif 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]).6

Theorem A.74 (Spectrum). Let T (L; L) be a compact operator, then ∈L (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 as- cent, with the property∈ that of ker(µI T )α = ker(µI T )α+1. Then dim ker(µI T )α is called the algebraic multiplicity− of µ and− dim ker(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) be a compact operator. Then ∞ ∈L (i) 0 σ(T ). ∈ 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). The operator T is said to be symmetric if (Tv,w) = (v,Tw) for∈L all v,w H; equivalently, iden- H H ∈ tifying 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) be a symmetric operator. Then, σ(T ) R and ∈L ⊂ a,b σ(T ) [a,b], (A.39) { }⊂ ⊂ with a = infv H, v H =1(Tv,v)H and b = supv H, v =1(Tv,v)H . Moreover, ∈ k k ∈ k kH T (L;L) = σ(T ) = max( a , b ). k kL | | | | | | 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) be a symmetric operator. Then λ σ(T ) if and only if there is ∈L ∈ a sequence (vn)n N in L such that vn L = 1 for all n N and Tvn λvn 0 as n . ∈ k k ∈ − → → ∞ 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) be a symmetric compact operator. Then L has a Hilbertian basis composed∈ L of eigenvectors of T .