Part XII, Chapter A

Banach and Hilbert Spaces

The goal of this appendix is to recall fundamental results on Banach andHilbert spaces. The results collected herein provide a theoretical frameworkfor the mathematical analysis of the finite element method. Some classicalresults 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 bijectiveoperators in Banach spaces. To get started, let us recall the following definitionof injective, surjective, and bijective maps.

Definition A.1 (Injection, surjection, bijection). Let E and F be twosets. A function (or map) f : E −→ F is said to be injective if every elementof 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 codomainis mapped to by at least one element of the domain. Finally, f is bijective ifevery element of the codomain is mapped to by exactly one element of thedomain (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 7−→ ‖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 .

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 saidto 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 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 vectorspace 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 normedvector spaces. L(V ;W ) is the vector space of bounded linear maps from V toW . 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 normedvector 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 Wis continuous. We say that V is continuously embedded into W and we writeV →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 letW be a Banach space. Equip L(V ;W ) with the norm

‖A‖L(V ;W ) = supv∈V

‖A(v)‖W‖v‖V

, ∀A ∈ L(V ;W ). (A.3)

Then, L(V ;W ) is a Banach space.

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-

tation by writing supv∈V‖A(v)‖W

‖v‖V instead of supv∈V \{0}‖A(v)‖W

‖v‖V . ⊓⊔

The Uniform Boundedness Principle (or Banach–Steinhaus Theorem) is auseful tool to study the limit of a sequence of operators in Banach spaces.

Theorem A.11 (Uniform Boundedness Principle). Let V and W be twoBanach spaces. Let {Ai}i∈I be a family (not necessarily countable) of operatorsin L(V ;W ). Assume that

supi∈I‖Aiv‖W <∞, ∀v ∈ V. (A.4)

Then, there is a constant C such that

‖Aiv‖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 Banachspaces. Let (An)n∈N be a sequence of operators in L(V ;W ) such that, for allv ∈ V , the sequence (Anv)n∈N converges as n → ∞ to a limit in W denotedAv (this means that the sequence (An)n∈N converges pointwise to A). Then,the following holds:

(i) supn∈N ‖An‖L(V ;W ) <∞;(ii) A ∈ L(V ;W );(iii) ‖A‖L(V ;W ) ≤ lim infn→∞ ‖An‖L(V ;W ).

Proof. Statement (i) is just a consequence of Theorem A.11. Owing to (A.5),we infer that ‖Anv‖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 inferthat statement (ii) holds. Finally, statement (iii) results from the fact that‖Anv‖W ≤ ‖An‖L(V ;W )‖v‖V for all v ∈ V and all n ∈ N. ⊓⊔

Remark A.13 (Uniform convergence on compact sets). Note thatCorollary A.12 does not claim that (An)n∈N converges to A in L(V ;W ),i.e., uniformly on bounded sets. However, a standard argument shows that(An)n∈N converges uniformly to A on compact sets. Indeed, let K ⊂ V be acompact set. Let ǫ > 0. Set C := supn∈N ‖An‖L(V ;W ); this quantity is finiteowing to statement (i) in Corollary A.12. K being compact, we infer thatthere is a finite set of points {xi}i∈I in K such that, for all v ∈ K, there isi ∈ I such that ‖v − xi‖V ≤ (6C)−1ǫ. Owing to the pointwise convergence of(An)n∈N to A, there is Ni such that, for all n ≥ Ni, ‖Anxi − Axi‖W ≤ 1

3ǫ.Using the triangle inequality and statement (iii) above, we infer that

‖Anv −Av‖W ≤ ‖An(v − xi)‖W + ‖Anxi −Axi‖W + ‖A(v − xi)‖W ≤ ǫ,

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 Banachspaces. T ∈ L(V ;W ) is called a compact operator if from every boundedsequence (vn)n∈N in V , one can extract a subsequence (vnk)k∈N such that thesequence (Tvnk)k∈N converges in W ; equivalently, T maps the unit ball in Vinto 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 casewhere V and W are two Banach spaces such that the injection of V intoW is compact. Then from every bounded sequence (vn)n∈N in V , one canextract a subsequence that converges in W . ⊓⊔

A.2.2 Duality

We start with real vector spaces and then discuss the extension to complexvector spaces.

Definition A.17 (Dual space, Bounded linear forms). Let V be anormed 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. Itsaction on an element v ∈ V is either denoted A(v) (or Av) or by means ofduality 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 thenorm

‖A‖V ′ = supv∈V

|A(v)|‖v‖V

= supv∈V

|〈A, v〉V ′,V |‖v‖V

, ∀A ∈ V ′. (A.6)

Note that the absolute value can be omitted from the numerators since A islinear and R-valued, and ±v can be considered in the supremum.

Theorem A.18 (Hahn–Banach). Let V be a normed vector space over Rand let W be a subspace of V . Let B ∈ W ′ = L(W ;R) be a bounded linear

map with norm ‖B‖W ′ = supw∈WB(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. ⊓⊔

Part XII. Appendices 777

Corollary A.19 (Dual characterization of norm). Let V be a normedvector space over R. Then, the following holds:

‖v‖V = supA∈V ′,‖A‖V ′=1

A(v) = supA∈V ′,‖A‖V ′=1

〈A, v〉V ′V , (A.7)

for all v ∈ V , and the supremum is attained.

Proof. Assume v 6= 0 (the assertion is obvious for v = 0). We first observe thatsupA∈V ′,‖A‖V ′=1A(v) ≤ ‖v‖V . Let W = span(v) and let B ∈ W ′ be definedas B(tv) = t‖v‖V for all t ∈ R. Owing to the Hahn–Banach Theorem, thereexists 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 spaceover R and let W be a subspace of V . Assume that any bounded linear formin 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 vectorspaces 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 overR is the dual of V ′ and is denoted V ′′.

Proposition A.23 (Isometric embedding into V ′′). Let V be a Banachspace over R. Then, V ′′ is a Banach space, and the linear map JV : V → V ′′

defined by

〈JV v, w′〉V ′′,V ′ = 〈w′, v〉V ′,V , ∀(v, w′) ∈ V × V ′, (A.9)

is an isometry.

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

‖JV v‖V ′′ = supw′∈V ′

‖w′‖V ′=1

|〈JV v, w′〉V ′′,V ′ | = supw′∈V ′

‖w′‖V ′=1

|〈w′, v〉V ′,V | = ‖v‖V ,

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 JV (V ) ⊂ V ′′. It mayhappen that the map JV is not surjective. In this case, the space V is aproper subspace of V ′′. For instance, L∞(D) = L1(D)′ but L1(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 spaceover R. V is said to be reflexive if JV is an isomorphism.

Let now V be a normed vector space over C. The notion of dual space ofV can be defined as in Definition A.17 by setting V ′ = L(V ;C). However, inthe context of weak formulations of PDEs with complex-valued functions, it ismore convenient to work with maps A : V → C that are antilinear ; this meansthat A(v + w) = Av +Aw for all v, w ∈ V (as usual), but A(λv) = λv for allλ ∈ C and all v ∈ V , where λ denotes the complex conjugate of λ (insteadof A(λv) = λv, in which case the map is linear). We denote by V ′ the vectorspace 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 ofthe elements of V by the action of the elements of V ′. To this purpose, it isuseful to consider V also as a vector space over R by restricting the scalingλv to λ ∈ R and v ∈ V . The corresponding vector space is denoted VR todistinguish it from V (thus, V and VR are the same sets, but equipped withdifferent structures). For instance, if V = Cm so that dim(V ) = m, thendim(VR) = 2m; a basis 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≤mwith i2 = −1. Another example is V = L2(0, 2π;C) for which an Hilbertianbasis is the set {cos(nx), sin((n+1)x)}n∈N, while an Hilbertian basis of VR isthe set {cos(nx), i cos(nx), sin((n+ 1)x), i sin((n+ 1)x)}n∈N.

Let V ′R 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 7→ 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 sothat A(v) = 0. Let us now prove that I is surjective. Let ψ ∈ V ′

R and considerthe 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 mapA : V → C is antilinear; indeed, for all λ ∈ C, writing λ = µ + iν withµ, 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

that ‖A‖V ′ ≤ ‖ψ‖V ′R. Let v ∈ V be such that A(v) 6= 0 and set λ = A(v)

|A(v)| ∈ C.Then,

|A(v)| = λ−1A(v) = A(λ−1v) = ψ(λ−1v) + iψ(iλ−1v),

but since ψ takes values in R, we infer that ψ(iλ−1v) = 0, so that |A(v)| =ψ(λ−1v). As a result,

|A(v)| ≤ ‖ψ‖V ′R‖λ−1v‖V = ‖ψ‖V ′

R‖v‖V ,

since |λ| = 1. This concludes the proof. ⊓⊔

Corollary A.27 (Dual characterization of norm). Let V be a normedvector space over C. Then, the following holds:

‖v‖V = maxA∈V ′,‖A‖V ′=1

ℜ(A(v)). (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 = maxA∈V ′,‖A‖V ′=1 |A(v)| since it is always possible to multiply Ain the supremum by a unitary 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 aboveconstruction; see Lax [321, p. 27]. ⊓⊔

The rest of the material is adapted straightforwardly. The adjoint of anoperator A ∈ L(V ;W ) is still defined by (A.8), and one can verify that itmaps (linearly) bounded antilinear maps in W ′ to bounded antilinear mapsin V ′. Moreover, the bidual is defined by considering bounded antilinear formson V ′, and the linear isometry extending that from Proposition A.23 is suchthat 〈JV 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 knownresults so as to derive new results that could difficult to obtain directly. Animportant application is the derivation of interpolation error estimates infractional-order Sobolev spaces. There are many interpolation methods; see,

780 Appendix A. Banach and Hilbert Spaces

e.g., Bergh and Lofstrom [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 intoa common topological vector space V. Then, V0+V1 is a normed vector spacewith the (canonical) norm ‖v‖V0+V1

= infv=v0+v1(‖v0‖V0+‖v1‖V1

). Moreover,if V0 and V1 are Banach spaces, then V0 + V1 is also a Banach space; see [47,Lem. 2.3.1]. For all v ∈ V0 + V1 and all t > 0, define

K(t, v) = infv=v0+v1

(‖v0‖V0+ t‖v1‖V1

). (A.11)

For all t > 0, v 7→ K(t, v) defines a norm on V0+V1 equivalent to the canonicalnorm. One can also verify that t 7→ K(t, v) is nondecreasing and concave (andtherefore continuous) and that t 7→ 1

tK(t, v) is increasing.

Definition A.30 (Interpolated space). Let θ ∈ (0, 1) and let p ∈ [1,∞].The interpolated space [V0, V1]θ,p is defined to be

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

where ‖ϕ‖Lp(R+; dtt ) =

(∫∞0|ϕ(t)|p dt

t

) 1p for p ∈ [1,∞) and ‖ϕ‖L∞(R+; dt

t ) =

sup0<t<∞ |ϕ(t)|. This space is equipped with the norm

‖v‖[V0,V1]θ,p = ‖t−θK(t, v)‖Lp(R+; dtt ).

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

[V0, V1]θ,p reduces to {0} if t−θmin(1, t) 6∈ Lp(R+;dtt ). In particular, [V0, V1]θ,p

is trivial if θ ∈ {0, 1} and p <∞. ⊓⊔

Remark A.32 (Gagliardo set). The map t 7→ K(t, v) has a simple geomet-ric interpretation. Introducing the Gagliardo set G(v) = {(x0, x1) ∈ R2 | v =v0+v1 with ‖v0‖V0

≤ x0 and ‖v1‖V1≤ x1}, one can verify that G(v) is convex

and that K(t, v) = infv∈∂G(v)(x0 + tx1), so that the map t 7→ K(t, v) is oneway to explore the boundary of G(v); see [47, p. 39]. ⊓⊔

Remark A.33 (Intersection). The vector space V0 ∩ V1 can be equippedwith the (canonical) norm ‖v‖V0∩V1

= max(‖v‖V0, ‖v‖V1

). For all v ∈ V0 ∩V1,one can verify that K(t, v) ≤ min(1, t)‖v‖V0∩V1

, whence we infer the contin-uous embedding V0 ∩ V1 → [V0, V1]θ,p for all θ ∈ (0, 1) and p ∈ [1,∞]. As aresult, if V0 ⊂ V1, then V0 → [V0, V1]θ,p. ⊓⊔

Lemma A.34 (Continuous embedding). Let θ ∈ (0, 1) and p, q ∈ [1,∞]with p ≤ q. Then, [V0, V1]θ,p → [V0, V1]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 toW0 and W1. Then, for all θ ∈ (0, 1) and all p ∈ [1,∞], A maps [V0, V1]θ,pboundedly to [W0,W1]θ,p. Moreover,

‖A‖L([V0,V1]θ,p;[W0,W1]θ,p) ≤ ‖A‖1−θL(V0;W0)‖A‖θL(V1;W1)

. (A.13)

Proof. See [443, Lem. 22.3]. ⊓⊔

Theorem A.36 (Lions–Peetre, reiteration). Let θ0, θ1 ∈ [0, 1] with(JLG) I changedθ0, θ1 ∈ (0, 1) toθ0, θ1 ∈ [0, 1] sincethis is what we needin general.

θ0 6= θ1. Assume that [V0, V1]θ0,1 → W0 → [V0, V1]θ0,∞ and [V0, V1]θ1,1 →W1 → [V0, V1]θ1,∞. Then, for all θ ∈ (0, 1) and all p ∈ [1,∞], [W0,W1]θ,p =[V0, V1]η,p with equivalent norms where η = (1− θ)θ0 + θθ1.

Proof. See Tartar [443, Thm. 26.2]. ⊓⊔

Theorem A.37 (Lions–Peetre, extension). Let V0, V1, F be three Banachspaces. Let A ∈ L(V0∩V1;F ), then A extends into a linear continuous mappingfrom [V0, V1]θ,1;J to F if and only if

∃c <∞, ∀v ∈ V0 ∩ V1 ‖Av‖F ≤ c‖v‖1−θV0‖v‖θV1

. (A.14)

Proof. See [443, Lem. 25.3]. ⊓⊔

Theorem A.38 (Interpolation of dual spaces). Let θ ∈ (0, 1) and p ∈[1,∞). Then, [V0, V1]

′θ,p = [V ′

1 , V′0 ]1−θ,p′ where p

′ = pp−1 (with the convention

that p′ =∞ if p = 1).

Proof. See Bergh and Lofstrom [47].page! ⊓⊔

A.3 Hilbert spaces

We start with real vector spaces and then briefly discuss the extension tocomplex vector spaces.

Definition A.39 (Inner product). Let V be a vector space over R. An innerproduct (or scalar product) on V is a map

(·, ·)V : V × V ∋ (v, w) 7−→ (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 itis 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.

782 Appendix A. Banach and Hilbert Spaces

Proposition A.40 (Cauchy–Schwarz). Let (·, ·)V be an inner product onthe real vector space V . By setting

‖v‖V = (v, v)12

V , ∀v ∈ V, (A.16)

one defines a norm on V . Moreover, the Cauchy–Schwarz1 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

consequence of the identity ‖v‖V ‖w‖V −(v, w)V = ‖v‖V ‖w‖V2

∥∥ v‖v‖V −

w‖w‖V

∥∥2V,

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 x1, . . . , xn benon-negative numbers. Then,

(x1x2 . . . xn)1n ≤ 1

n (x1 + . . .+ xn). (A.18)

Proof. Use the convexity of the function x 7→ ex. ⊓⊔

This inequality is frequently used in conjunction with the Cauchy–Schwarzinequality. In particular, it implies that

|(v, w)V | ≤ γ2 ‖v‖2V + 1

2γ ‖w‖2V , ∀γ > 0, ∀v, w ∈ V. (A.19)

Definition A.43 (Hilbert spaces). A Hilbert space V is an inner productspace over R that is complete with respect to the induced norm (and is, there-fore, a Banach space). The inner product is denoted (·, ·)V and the inducednorm ‖·‖V . A Hilbert space is said to be separable if it admits a countable anddense subset.

Theorem A.44 (Riesz–Frechet). Let V be a Hilbert space over R. For eachv′ ∈ 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 ′ 7→ 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–Frechet Theorem is the following:

Proposition A.45 (Reflexivity). Hilbert spaces are reflexive.

Proof. Let V be a Hilbert space. The Riesz–Frechet Theorem implies that Vcan 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 overR. 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 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 char-

acterized by the condition (f − u, v)V = 0 for all v ∈ U , and the mapΠU : V ∋ f 7→ u ∈ U is linear with ‖ΠU‖L(V ;U) = 1 (unless U = {0} sothat ‖ΠU‖L(V ;U) = 0).

Proof. See Exercise 17.3. ⊓⊔

Let now V be a vector space over C. Then, an inner product on V is amap (·, ·)V : V × V ∋ (v, w) 7−→ (v, w)V ∈ C satisfying the following threeproperties:

(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)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 7→ (v, v)1/2V still defines a norm on V , and the Cauchy–

Schwarz inequality still takes the form (A.17) (with the modulus on the left-hand side). The Riesz–Frechet Theorem A.44 still states that, for all v′ ∈ V ′

(recall that v′ is antilinear by convention), there exists a unique u ∈ V suchthat 〈v′, w〉V ′,V = (u,w)V for all w ∈ V , and the map v′ ∈ V ′ 7→ u ∈ Vis an isometric (linear) isomorphism. Finally, the Euler–Lagrange conditionfrom Proposition A.46 now becomes ℜ(f − u, v − u)V ≤ 0 for all v ∈ U . Theproof of these extensions hinges on the fact that, if V is a complex Hilbertspace equipped with the inner product (·, ·)V , then VR equipped with the innerproduct ℜ(·, ·)V is a real Hilbert space (recall that V and VR are the samesets, equipped with different structures). Let us prove for instance the Riesz–Frechet Theorem. Let v′ ∈ V ′. Using the bijective isometry from Lemma A.26,we consider I(v′) ∈ V ′

R. Owing to Theorem A.44 on VR equipped with ℜ(·, ·)V ,there is a unique u ∈ VR such that ℜ(u,w)V = I(v′)(w) = ℜ〈v′, w〉V ′,V for allw ∈ VR. 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 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 andQuartapelle [264]. For simplicity, we implicitly assume that V and W are realBanach 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. Theoperator A being bounded, ker(A) is closed in V . Hence, the quotient of Vby ker(A), V/ker(A), can be defined. This space is composed of equivalenceclasses v such that v and w are in the same class v if and only if v−w ∈ ker(A).

Theorem A.47 (Quotient space). The space V/ker(A) is a Banach spacewhen equipped with the norm ‖v‖ = infv∈v ‖v‖V . Moreover, defining A :V/ker(A)→ im(A) by Av = Av for all v in v, A is an isomorphism.

Proof. See Brezis [97, §11.2], Yosida [483, p. 60]. ⊓⊔

For subspacesM ⊂ V and N ⊂ V ′, we introduce the so-called annihilatorsof M and N which are defined as follows:

M⊥ = {v′ ∈ V ′ | ∀m ∈M, 〈v′,m〉V ′,V = 0}, (A.21)

N⊥ = {v ∈ V | ∀n′ ∈ N, 〈n′, v〉V ′,V = 0}. (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 prop-erties 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. 202-209]. ⊓⊔

Showing that the range of an operator is closed is a crucial step towardsproving that this operator is surjective. This is the purpose of the followingfundamental theorem:

Theorem A.49 (Banach or Closed Range). Let A ∈ L(V ;W ). The fol-lowing 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]. ⊓⊔

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 isan 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 par-ticular, 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), ∃vw ∈ V, Avw = w and α‖vw‖V ≤ ‖w‖W . (A.23)

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 = BV (0, 1)(the open unit ball in V ) yields that A(BV (0, 1)) is open in im(A). Since0 ∈ A(BV (0, 1)), there is γ > 0 such that BW (0, γ) ⊂ A(BV (0, 1)). Letw ∈ im(A). Since γ

2w

‖w‖W ∈ BW (0, α), there is z ∈ BV (0, 1) such that

Az = γ2

w‖w‖W . Setting v = 2‖w‖W

γ z leads Av = w and γ2 ‖v‖V ≤ ‖w‖W .

(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 thatAvn = wn and α‖vn‖V ≤ ‖wn‖W . Then, (vn)n∈N is a Cauchy sequence in V .Since V is a Banach space, (vn)n∈N converges to a certain v ∈ V . Owing tothe 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.51is 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−1w‖V ≤1α‖w‖W , i.e., A−1 is bounded. ⊓⊔

Let us finally give a sufficient condition for the image of an injective op-erator to be closed.

Lemma A.53 (Peetre–Tartar). Let X, Y , Z be three Banach spaces. LetA ∈ L(X;Y ) be an injective operator and let T ∈ L(X;Z) be a compactoperator. If there is c > 0 such that c‖x‖X ≤ ‖Ax‖Y + ‖Tx‖Z , then im(A) isclosed; equivalently, there is α > 0 such that

∀x ∈ X, α‖x‖X ≤ ‖Ax‖Y . (A.24)

786 Appendix A. Banach and Hilbert Spaces

Proof. By contradiction. Assume that there is a sequence (xn)n∈N of X suchthat ‖xn‖X = 1 and ‖Axn‖Y converges to zero when n goes to infinity. SinceT 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‖xnk − xmk‖X ≤ ‖Axnk −Axmk‖Y + ‖Txnk − Txmk‖Z ,

(xnk)k∈N is a Cauchy sequence in X. Let x be its limit. Clearly, ‖x‖X = 1.The boundedness of A implies Axnk → Ax and Ax = 0 since Axnk → 0. SinceA 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 MappingTheorem, we deduce two lemmas characterizing surjective operators.

Lemma A.54 (Surjectivity of A∗). Let A ∈ L(V ;W ). The following state-ments 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

infv∈V

supw′∈W ′

〈w′, Av〉W ′,W

‖w′‖W ′‖v‖V≥ α. (A.26)

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′

v′ ∈ W ′ such that A∗w′v′ = v′ and ‖w′

v′‖W ′ ≤ α−1‖v′‖V ′ . Let nowv ∈ 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 supw′∈W ′

〈w′, Av〉W ′,W

‖w′‖W ′.

Taking the supremum in v′ ∈ V ′ yields (A.26) since

‖v‖V = supv′∈V ′

〈v′, v〉V ′,V

‖v′‖V ′≤ α−1 sup

w′∈W ′

〈w′, Av〉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 (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) isclosed.(ii) ⇒ (i). Since im(A) is closed, we use Theorem A.49(iv) together with theinjectivity of A to infer that im(A∗) = {0}⊥ = V ′. ⊓⊔

Lemma A.55 (Surjectivity of A). Let A∈L(V ;W ). The following state-ments 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

infw′∈W ′

supv∈V

〈A∗w′, v〉V ′,V

‖w′‖W ′‖v‖V≥ α. (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.55is 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 = fhas 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 determinewhether the constant α in (A.28) is the same as that in (A.23). The answer tothis 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 spacesand let A ∈ L(V ;W ) be a surjective operator. Let α > 0. The property

∀w ∈W, ∃vw ∈ V, Avw = w and α‖vw‖V ≤ ‖w‖W , (A.29)

implies

infw′∈W ′

supv∈V

〈A∗w′, v〉V ′,V

‖w′‖W ′‖v‖V≥ α. (A.30)

The converse is true if V is reflexive. In the complex case, it is the real partof 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 ′ = supw∈W

‖w‖W=1

〈w′, w〉W ′,W .

For all w in W , there is vw ∈ V such that Avw = w and α‖vw‖V ≤ ‖w‖W .Let w′ in W ′. Therefore,

〈w′, w〉W ′,W = 〈w′, Avw〉W ′,W = 〈A∗w′, vw〉V ′,V ≤ 1α‖A∗w′‖V ′‖w‖W .

Hence,‖w′‖W ′ = sup

w∈W‖w‖W=1

〈w′, w〉W ′,W ≤ 1α‖A∗w′‖V ′ .

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′‖V ′ ≥ α‖w′‖W ′ forall w′ ∈W ′, A∗ is injective. Let v′ ∈ im(A∗) and define z′(v′) ∈W ′ such thatA∗(z′(v′)) = v′. Note that z′(v′) is unique since A∗ is injective; this in turnimplies that z′(·) : im(A∗) ⊂ V ′ → W ′ is a linear mapping. (Note also thatz′ 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 impliesz′(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 vw, satisfying (A.23). We first define the linear form φw : im(A∗)→ R by

∀v′ ∈ im(A∗), φw(v′) = 〈z′(v′), w〉W ′,W ,

i.e., φw(A∗w′) = 〈w′, w〉W ′,W for all w′ ∈W ′. Hence,

|φw(v′)| ≤ ‖z′(v′)‖W ′‖w‖W ≤ 1α‖A∗z′(v′)‖V ′‖w‖W ≤ 1

α‖v′‖V ′‖w‖W .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 thesame norm. Let φw ∈ V ′′ be the extension in question with ‖φw‖V ′′ ≤ 1

α‖w‖W .

Since V is assumed to be reflexive, there is vw ∈ V such that JV (vw) = φw.As a result,

∀w′ ∈W ′, 〈w′, Avw〉W ′,W = 〈A∗w′, vw〉V ′,V = 〈JV (vw), A∗w′〉V ′′,V ′

= 〈φw, A∗w′〉V ′′,V ′ = φw(A∗w′)

= 〈z′(A∗w′), w〉W ′,W = 〈w′, w〉W ′,W ,

showing that Avw = w. Hence, vw is an inverse image of w and

‖vw‖V = ‖JV (vw)‖V ′′ = ‖φw‖V ′′ ≤ 1α‖w‖W . ⊓⊔

Remark A.58 (Linearity). Note that the dependence of vw with respectto w in Lemma A.57 is a priori nonlinear. Linearity can be obtained whenthe setting is Hilbertian by using the zero extension of φw on the orthogonalcomplement 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 ∈ L(V ;W ). The following state-ments 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

‖Av‖W ≥ α‖v‖V , ∀v ∈ V, (A.31)

or, equivalently, A∗ is injective and

infv∈V

supw′∈W ′

〈w′, Av〉W ′,W

‖w′‖W ′‖v‖V=: α > 0. (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 equivalentto 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 isinjective, this yields (ii). Finally, to prove that (ii) implies (i), we only needto prove that (ii) implies the surjectivity of A. The injectivity of A∗ impliesim(A) = (ker(A∗))⊥ = W . Since im(A) is closed, im(A) = W , i.e., A issurjective. ⊓⊔

Remark A.60 (Bijectivity of A∗). The bijectivity of A ∈ L(V ;W ) isequivalent to that of A∗ ∈ L(W ′;V ′). Indeed, statement (ii) in Theorem A.59is equivalent to A∗ injective and A∗ surjective owing to the equivalence ofstatements (i) and (ii) from Lemma A.54. ⊓⊔

Corollary A.61 (Inf-sup condition). Let A ∈ L(V ;W ) be a bijective op-erator. Assume that V is reflexive. Then,

infv∈V

supw′∈W ′

〈w′, Av〉W ′,W

‖w′‖W ′‖v‖V= infw′∈W ′

supv∈V

〈w′, Av〉W ′,W

‖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 twopositive finite numbers, since A is a bijective bounded operator. The left-hand 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 vw ∈ V so that Avw = w and the previous statement regarding limplies that l ‖vw‖V ≤ ‖w‖W . This in turn implies that

790 Appendix A. Banach and Hilbert Spaces

‖w′‖W ′ = supw∈W

〈w′, w〉W ′,W

‖w‖W= supw∈W

〈w′, Avw〉W ′,W

‖w‖W= supw∈W

〈A∗w′, vw〉V ′,V

‖w‖W

≤ ‖A∗w′‖V ′ supw∈W

‖vw‖V‖w‖W

≤ 1

l‖A∗w′‖V ′ ,

which implies l ≤ r. That r ≤ l is proved similarly by working with W ′ in lieuof V , V ′ in lieu of W and A∗ in lieu of A. The above reasoning leads to

infw′∈W ′

supv′′∈V ′′

〈v′′, A∗w′〉V ′′,V ′

‖v′′‖V ′′‖w′‖W ′≤ infv′′∈V ′′

supw′∈W ′

〈v′′, A∗w′〉V ′′,V ′

‖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 6=0 is not bijective. For instance if A : (x1, x2, x3 . . .) 7−→ (0, x1, x2, x3, . . .) isthe right shift operator in ℓ2, then A∗ : (x1, x2, x3 . . .) 7−→ (x2, x3, x4, . . .) isthe left shift operator. It can be verified that A is injective but not surjectivewhereas A∗ is injective but not surjective. It can also be shown that l = 1 andr = 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 > 0and ξ = ±1 such that

ξ〈Av, v〉V ′,V ≥ α‖v‖2V , ∀v ∈ V. (A.34)

In the complex case, A ∈ L(V ;V ′) is said to be a coercive operator if thereexist a real number α > 0 and a complex number ξ with |ξ| = 1 such that

ℜ (ξ〈Av, v〉V ′,V ) ≥ α‖v‖2V , ∀v ∈ V. (A.35)

The following proposition shows that the notion of coercivity is relevantonly in Hilbert spaces:

Proposition A.64 (Hilbert structure). Let V be a Banach space. V canbe equipped with a Hilbert structure with the same topology if and only if thereis a coercive operator in L(V ;V ′).

Proof. See Exercise 17.5. ⊓⊔

Corollary A.65 (Sufficient condition). Coercivity is a sufficient conditionfor an operator A ∈ L(V ;V ′) to be bijective.

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 Banachspace, so that V and V ′′ are identified. The operator A ∈ L(V ;V ′) is saidto be self-adjoint if A∗ = A in the real case and if A∗ = A in the complexcase.

Self-adjoint bijective operators are characterized as follows:

Corollary A.67 (Self-adjoint bijective operator). Let V be a reflexiveBanach space and let A ∈ L(V ;V ′) be a self-adjoint operator. Then, A isbijective 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 satisfiesinequality (A.36). Conversely, inequality (A.36) means that A is injective. Itfollows that A∗ is injective since A∗ = A (or A∗ = A) by hypothesis. Theconclusion 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 Banachspace and let A ∈ L(V ;V ′) be a monotone self-adjoint operator. Then, Ais bijective if and only if A is coercive (with ξ = 1).

Proof. See Exercise 17.6. ⊓⊔The rest of thischapter still to bechecked

A.5 Spectral theory

We briefly recall in this section some essential facts regarding the spectraltheory 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 spaceand let T ∈ L(L;L). The resolvent set, ρ(T ), and the spectrum of T , σ(T ),are the sets in C such that

792 Appendix A. Banach and Hilbert Spaces

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

σ(T ) = C\ρ(T ). (A.38)

µ ∈ C is called eigenvalue of T if ker(µI − T ) 6= {0} and ker(µI − T ) is theassociated 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 )| = limn→∞ ‖Tn‖1n

L(L;L) where |σ(T )| := maxλ∈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 Banachspaces is compact if and only if its adjoint is.

Theorem A.73 (Fredholm alternative). Let T ∈ L(L;L) be a compactoperator 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 equivalentway: Either µI−T is bijective or ker(µI−T ) 6= 0. The key result for compactoperators 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 as-

cent, with the property that of ker(µI − T )α = ker(µI − T )α+1. Thendimker(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 thosein ker(µI − T )α are called generalized eigenvectors. Both ker(µI − T )α andker(µI − T ) are invariant under T . Note that the ascent of µI − T is one andthe two multiplicities are equal if T is self-adjoint; in this case the eigenvaluesare 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 compactoperator. Then

(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;L). The operator T is saidto be symmetric if (Tv,w)H = (v, Tw)H for all v, w ∈ H; equivalently, iden-tifying L and L′, T is self-adjoint, i.e., T = T ∗. The key result for symmetricoperators is the following (see, e.g., [321, p. 356]).

Theorem A.76 (Real spectrum, spectral radius). Let L be a Hilbertspace 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 = infv∈H,‖v‖H=1(Tv, v)H and b = supv∈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 andlet T ∈ L(L;L) be a symmetric operator. Then λ ∈ σ(T ) if and only if there isa sequence (vn)n∈N in L such that ‖vn‖L = 1 for all n ∈ N and Tvn−λvn → 0as n→∞.

We conclude this section by considering symmetric compact operators.

Proposition A.78 (Symmetric compact operator). Let L be a Hilbertspace and let T ∈ L(L;L) be a symmetric compact operator. Then L has aHilbertian basis composed of eigenvectors of T .