Appendix a Linear Operators in Banach Spaces

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Appendix A Linear operators in Banach Spaces

The goal of this appendix is to recall fundamental results on linear operators in Banach and Hilbert spaces; see, e.g., Aubin [15], Brezis [56], Lax [166], Rudin [206], Yosida [251], Zeidler [252] for further reading. The results collected in this appendix provide a theoretical framework for the mathematical analysis of the ﬁnite element method.

A.1 Banach and Hilbert spaces

To stay general, we consider complex vector spaces, i.e., vector spaces over the ﬁeld C of complex numbers. The case of real vector spaces is recovered by replacing the ﬁeld C by R, by removing the real part symbol ( ) and the complex conjugate symbol , and by interpreting the symbol asℜ the· absolute value instead of the modulus.· Some additional minor adaptations|·| (regarding duality and inner products) are indicated whenever relevant.

A.1.1 Banach spaces Let V be a complex vector space.

Definition A.1 (Norm). A norm on V is a map V : V [0, ) satisfying the following three properties: k·k → ∞ (i) Definiteness: v =0 v =0. k kV ⇐⇒ (ii) 1-homogeneity: λv V = λ v V , for all λ C and all v V . (iii) Triangle inequality:k k v + |w|k k v + w ∈ , for all v, w∈ V . k kV ≤k kV k kV ∈ A seminorm on V is a map : V [0, ) satisfying only (ii) and (iii). |·|V → ∞ Definition A.2 (Banach space). Equip V with a norm V ; then V is called a Banach space if every Cauchy sequence (with respectk·k to the metric d(x, y)= x y ) in V has a limit in V . k − kV 424 Appendix A. Linear operators in Banach Spaces

Deﬁnition A.3 (Equivalent norms). Two norms V,1 and V,2 are said to be equivalent on V if there exists a positive real numberk·k c suchk·k that

1 c v v c− v , v V. (A.1) k kV,2 ≤k kV,1 ≤ k kV,2 ∀ ∈ One readily verifies that when (A.1) holds, V is a Banach space for the norm iff it is a Banach space for the norm . k·kV,1 k·kV,2 Remark A.4 (Finite dimension). If the vector space V is finite-dimensional, all the norms in V are equivalent. This result is false in infinite-dimensional vector spaces. Actually, the unit ball in V is a compact set (for the norm topology) if and only if V is finite-dimensional; see Brezis [56, Thm. 6.5], Lax [166, 5.2]. § ⊓⊔ A.1.2 Bounded linear maps Let V, W be complex vector spaces. A map A : V W is said to be linear → if A(v1 + v2) = A(v1)+ A(v2) for all v1, v2 V and A(λv) = λA(v) for all λ C and all v V . To alleviate the notation,∈ we simply write Av instead ∈ ∈ of A(v). Whenever V and W can be equipped with norms V and W , respectively, the linear map A is said to be bounded or continuousk·k if k·k

Av W A (V ;W ) := sup k k < . (A.2) k kL v V v V ∞ ∈ k k In this book, we systematically abuse the notation by implicitly assuming that the argument in this type of supremum or infimum is nonzero. Proposition A.5 ( (V ; W )). The complex vector space composed of the L bounded linear maps from V to W is denoted (V ; W ). The map (V ;W ) defined by (A.2) is indeed a norm on (V ; W ),L and if W is a Banachk·kL space, then (V ; W ) equipped with this normL is also a Banach space. L Proof. See Rudin [206, p. 87], Yosida [251, p. 111]. ⊓⊔ Example A.6 (Continuous embedding). Assume that V W and that ⊂ there is a real number c such that v W c v V for all v V . Then the embedding of V into W is continuous.k Wek say≤ thatk kV is continuously∈ embedded . into W and we write V ֒ W → ⊓⊔ Whenever V and W are Banach spaces, maps in (V ; W ) are often called operators. The following result (often called UniformL Boundedness Principle) is a useful tool to study the limit of a sequence of operators. Theorem A.7 (Banach–Steinhaus). Let V, W be Banach spaces and let Ai i I be a family (not necessarily countable) in (V ; W ). Assume that { } ∈ L supi I Aiv W is a finite number for all v V . Then, there is a real number C such∈ k thatk ∈ sup Aiv W C v V , v V. (A.3) i I k k ≤ k k ∀ ∈ ∈ Appendices 425

Proof. See Brezis [56, p. 32], Lax [166, Chap. 10]. ⊓⊔ Corollary A.8 (Pointwise convergence). Let V, W be Banach spaces. Let (An)n N be a sequence in (V ; W ) such that, for all v V , the sequence ∈ L ∈ (Anv)n N converges as n to a limit in W denoted Av (this means that ∈ → ∞ the sequence (An)n N converges pointwise to A). Then, the following holds: ∈ (i) supn N An (V ;W ) < ; (ii) A ∈ (Vk; Wk)L; ∞ ∈ L (iii) A (V ;W ) lim infn An (V ;W ). k kL ≤ →∞ k kL Proof. Statement (i) is just a consequence of Theorem A.7. Owing to (A.3), we infer that A v C v for all v V and all n N. Letting n , k n kW ≤ k kV ∈ ∈ → ∞ we obtain that Av W C v V , and since A is obviously linear, we infer that statement (ii)k holds.k ≤ Finally,k k statement (iii) results from the fact that Anv W An (V ;W ) v V for all v V and all n N. k k ≤k kL k k ∈ ∈ ⊓⊔ Remark A.9 (Uniform convergence on compact sets). Corollary A.8 does not claim that (An)n N converges to A in (V ; W ), i.e., uniformly on ∈ L bounded sets. However, a standard argument shows that (An)n N converges uniformly to A on compact sets. Indeed, let K V be a compact∈ set. Let ⊂ ǫ> 0. Set C := supn N An (V ;W ); C is finite owing to Corollary A.8(i). The ∈ k kL set K being compact, we infer that there is a finite set of points xi i I in K { 1} ∈ such that, for all v K, there is i I such that v x (3C)− ǫ. Owing ∈ ∈ k − ikV ≤ to the pointwise convergence of (An)n N to A, there is Ni such that, for all 1 ∈ n Ni, Anxi Axi W 3 ǫ. Using the triangle inequality and statement (iii)≥ above,k we infer− thatk ≤ A v Av A (v x ) + A x Ax + A(v x ) ǫ, k n − kW ≤k n − i kW k n i − ikW k − i kW ≤ for all v K and all n maxi I Ni. ∈ ≥ ∈ ⊓⊔ A.1.3 Compact operators Definition A.10 (Compact operator). Let V, W be Banach spaces. The operator T (V ; W ) is said to be compact if from every bounded sequence ∈ L (vn)n N in V one can extract a subsequence (vn )k N such that the sequence ∈ k ∈ (T vnk )k N converges in W ; equivalently, T maps the unit ball in V into a relatively∈ compact set in W . Theorem A.11 (Schauder). A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact. Example A.12 (Compact embedding). Assume that V W and that the embedding of V into W is compact. Then from every bounded⊂ sequence (vn)n N in V one can extract a subsequence that converges in W . ∈ ⊓⊔ Proposition A.13 (Composition). Let W , X, Y , Z be four Banach spaces and let A (Z; Y ), K (Y ; X), B (X; W ). Assume that K is compact; then the operator∈ L B K∈ LA is compact.∈ L ◦ ◦ 426 Appendix A. Linear operators in Banach Spaces

A.1.4 Duality Let V, W be complex vector spaces. The map A : V W is said to be → antilinear if A(v1 + v2)= Av1 + Av2 for all v1, v2 V and A(λv) = λAv for all λ C and all v V . The reason for considering∈ antilinear maps is that the weak∈ form of complex-valued∈ PDEs is derived by multiplying the PDE by the complex conjugate of a test function; therefore, the right-hand side of the weak formulation is an antilinear form. Alternatively, one could work with linear maps only upon noticing that if A is an antilinear map, then A (deﬁned by Av := Av for all v V ) is a linear map. ∈

Deﬁnition A.14 (Dual space). The dual space of V is denoted V ′ and is composed of the bounded antilinear forms from V to C. An element A V ′ is called bounded antilinear form, and its action on an element v V is denoted∈ ∈ either Av or A, v ′ . h iV ,V

Owing to Proposition A.5, V ′ is a Banach space with the norm

Av A, v V ′,V A V ′ = sup | | = sup |h i |, A V ′. (A.4) k k v V v V v V v V ∀ ∈ ∈ k k ∈ k k In the real case, the absolute value can be omitted from the numerators since v can be considered. In the complex case, the modulus can be replaced by ±the real part since v can be multiplied by any unit complex number.

Theorem A.15 (Hahn–Banach). Let V be a normed vector space over C and let W be a subspace of V . Let B W ′. There exists A V ′ that extends ∈ ∈ B, i.e., Aw = Bw for all w W , and such that A ′ = B ′ . ∈ k kV k kW Proof. For the real case, see Brezis [56, p. 3], Lax [166, Chap. 3], Rudin [206, p. 56], Yosida [251, p. 102]. The above statement is a simpliﬁed version of the actual Hahn–Banach Theorem. For the complex case, see Lax [166, p. 27], Brezis [56, Prop. 11.23]. ⊓⊔ Corollary A.16 (Norm by duality). The following holds:

Av A, v V ′V v V = sup | | = sup |h i |, (A.5) k k A V ′ A V ′ A V ′ A V ′ ∈ k k ∈ k k for all v V , and the supremum is attained. ∈ Proof. Assume v = 0 (the assertion is obvious for v = 0). We ﬁrst observe Av 6 that sup ′ | | v since Av A ′ v . Let W = span(v) and A V A ′ V V V ∈ k kV ≤ k k | |≤k k k k let B W ′ be deﬁned as B(λv) = λ v for all λ C; by construction, ∈ k kV ∈ B W ′ and B ′ = 1. Owing to the Hahn–Banach Theorem, there exists ∈ k kW A V ′ such that A ′ = 1 and Av = Bv = v . ∈ k kV k kV ⊓⊔ Appendices 427

Proposition A.17 (Double dual). The double dual of a Banach space V is denoted V ′′ and is deﬁned to be the dual space of its dual space V ′. Then the bounded linear map J : V V ′′ such that V →

J v, φ′ ′′ ′ = φ , v ′ , (v, φ′) V V ′, (A.6) h V iV ,V h ′ iV ,V ∀ ∈ × is an isometry.

Proof. The claim follows from Corollary A.16 since

JV v, φ′ V ′′,V ′ φ′, v V ′,V JV v V ′′ = sup |h i | = sup |h i | = v V . k k φ′ V ′ φ′ V ′ φ′ V ′ φ′ V ′ k k ⊓⊔ ∈ k k ∈ k k Definition A.18 (Reflexivity). A Banach space V is said to be reflexive if JV is an isomorphism.

Remark A.19 (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 1.1 or Brezis [56, 4.3]. § § ⊓⊔ Deﬁnition A.20 (Adjoint operator). (i) Let A (V ; W ). The adjoint ∈ L operator of A is the bounded linear operator A∗ (W ′; V ′) such that ∈ L

A∗w′, v ′ = w′, Av ′ , (v, w′) V W ′. (A.7) h iV ,V h iW ,W ∀ ∈ × ( ) (ii) Assume that W is reﬂexive. Let A (V ; W ′) and A ∗ := A∗ J ∈ L ◦ W ∈ (W ; V ′), then L ( ) A ∗ w, v ′ := Av, w ′ , (v, w) V W. (A.8) h iV ,V h iW ,W ∀ ∈ ×

Whenever the context is unambiguous, we identify W and W ′′ (when W is ( ) reﬂexive) and write A∗ instead of A ∗ .

Deﬁnition A.21 (Self-adjoint operator). Let V be a reﬂexive Banach ( ) space. The operator A (V ; V ′) is said to be self-adjoint if A = A ∗ , i.e., ∈ L

Av, w ′ = Aw, v ′ , v, w V. (A.9) h iV ,V h iV ,V ∀ ∈

In particular, Av, v ′ takes real values if A is self-adjoint. h iV ,V Remark A.22 (Hermitian transpose). In ﬁnite dimension, A can be represented by a matrix with complex-valued entries; then A∗ is represented by the Hermitian transpose of this matrix. A self-adjoint operator is represented by a Hermitian-symmetric matrix. ⊓⊔ 428 Appendix A. Linear operators in Banach Spaces

Remark A.23 (Space VR). Let V be a vector space over C. By restricting the scaling λv to λ R for all v V , V can also be equipped with a vector ∈ ∈ space structure over R, which we denote VR (V and VR are the same sets, but they are equipped with diﬀerent vector structures). For instance, if V = Cm then dim(V )= m but dim(VR)=2m; moreover, the canonical set ek 1 k m, m { } ≤ ≤ where the Cartesian components of ek in C are ek,l = δkl (the Kronecker symbol) for all l 1:m , is a basis of V , whereas the set ek,iek 1 k m 2 ∈ { } { } ≤ ≤ with i = 1 is a basis of VR. Let VR′ be the dual space of VR, i.e., the normed real vector− space composed of the bounded R-linear maps from V to R. Then the map I : V ′ VR′ such that, for all ℓ V ′, I(ℓ)(v)= (ℓ(v)) for all v V , is a bijective isometry;→ see [56, Prop. 11.22].∈ ℜ ∈ ⊓⊔ A.1.5 Hilbert spaces Let V be a complex vector space.

Definition A.24 (Inner product). An inner product on V is a map ( , )V : V V C satisfying the following three properties: (i) Sesquilinearity· · (the prefix× sesqui→ means one and a half): ( , w) is a linear map for fixed w V , · V ∈ whereas (v, )V is an antilinear map for fixed v V ; if V is a real vector space, then the· inner product is a bilinear map (i.e.,∈ it is linear in both of its arguments). (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 = 0 V ≥ ∈ V ⇐⇒ (note that (v, v)V is always real owing to Hermitian symmetry).

Deﬁnition A.25 (Hilbert space). A Hilbert space V is an inner product space that is complete with respect to the induced norm (and is, therefore, a Banach space). A Hilbert space is said to be separable if it admits a countable and dense subset.

Proposition A.26 (Cauchy–Schwarz). Let ( , )V be an inner product on V . By setting · · 1 v = (v, v) 2 , v V, (A.10) k kV V ∀ ∈ one deﬁnes a norm on V . Moreover, the Cauchy–Schwarz1 inequality holds:

(v, w) v w , v, w V. (A.11) | V |≤k kV k kV ∀ ∈ Remark A.27 (Equality). Equality holds in (A.11) iﬀ v and w are collinear. v V w V v w 2 This follows from v V w V (ξ(v, w)V )= k k k k ξ , for k k k k −ℜ 2 v V − w V V C k k k k all non-zero v, w V and all ξ with ξ = 1. ∈ ∈ | | ⊓⊔ Let x1,...,xn be non-negative real numbers. Then, using the convexity of the function x ex, one can show the following arithmetic-geometric inequality: 7→

1 Augustin-Louis Cauchy (1789–1857) and Herman Schwarz (1843–1921) Appendices 429

1 1 (x x ...x ) n (x + ... + x ). (A.12) 1 2 n ≤ n 1 n This inequality is frequently used in conjunction with the Cauchy–Schwarz inequality. In particular, it implies the following inequality, often called Young’s inequality: For any positive real number γ > 0,

(v, w) γ v 2 + 1 w 2 , v, w V. (A.13) | V |≤ 2 k kV 2γ k kV ∀ ∈

RF Theorem A.28 (Riesz–Fr´echet). Let V be a Hilbert space. The map JV : V V ′ such that → RF J v, w ′ = (v, w) , v, w V, (A.14) h V iV ,V V ∀ ∈ is a linear isometric isomorphism.

Proof. See Brezis [56, Thm. 5.5], Lax [166, p. 56], Yosida [251, p. 90]. ⊓⊔

The Riesz–Fréchet Theorem says that for all φ V ′, there exists a unique RF ∈ v V such that JV v = φ; the element v V is called the Riesz–Fréchet representative∈ of the bounded (anti)linear form∈ φ. Another simple, but important, consequence of this theorem is the following result. Corollary A.29 (Reflexivity). Hilbert spaces are reflexive.

Remark A.30 (Space VR). The reader is invited to verify that if V is a complex Hilbert space with inner product ( , )V , then VR is a real Hilbert space with inner product ( , ) . · · ℜ · · V ⊓⊔

A.2 Surjective and bijective operators

This section presents classical results to characterize surjective and bijective Banach operators, see Aubin [15], Brezis [56], Yosida [251].

Deﬁnition A.31 (Injection, surjection, bijection). Let E and G be two nonempty sets. A function (or map) f : E G is said to be injective if every element of the codomain (i.e., G) 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).

Definition A.32 (Left, right inverse). Let E and G be two nonempty sets. Given a map f : E G, we say that f ‡ : G E is a left-inverse of f iff → → (f ‡ f)(e)= e for all e E. We say that f † : G E is a right-inverse of f ◦ ∈ → iff (f f †)(g)= g for all g G. ◦ ∈ 430 Appendix A. Linear operators in Banach Spaces

A map with a left-inverse is necessarily injective. Conversely, if the map f : E G is injective, then (i) the map f˜ : E f(E) such that f˜(e) = f(e) for→ all e E has necessarily a unique left-inverse;→ (ii) one can then construct ∈ a left-inverse of f by setting f ‡(g) = e (with e E arbitrary) if g f(E) ∈ 6∈ and f ‡(g) = f˜‡(g) otherwise; (iii) if E, F are vector spaces and the map f is linear, the left-inverse of f˜ is also linear. A map with a right-inverse is necessarily surjective. Conversely, one can construct right-inverse maps for every surjective map upon invoking the axiom of choice.

A.2.1 Fundamental results Let V, W be Banach spaces. For A (V ; W ), we denote by ker(A) its kernel and by im(A) its range. The operator∈ L 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 classesv ˇ such that v and w are in the same class vˇ if and only if v w ker(A). − ∈ Theorem A.33 (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 [56, 11.2], Yosida [251, 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.15a) { ∈ | ∀ ∈ h iV ,V } N ⊥ = v V n′ N, n′, v ′ =0 . (A.15b) { ∈ | ∀ ∈ h iV ,V } Let M denote the closure of the subspace M in V . A characterization of ker(A) and im(A) is given by the following result. Lemma A.34 (Kernel and range). Let A (V ; W ). The following holds: ∈ L (i) ker(A) = (im(A∗))⊥. (ii) ker(A∗) = (im(A))⊥. (iii) im(A) = (ker(A∗))⊥. (iv) im(A ) (ker(A))⊥. ∗ ⊂ Proof. See Brezis [56, Cor. 2.18], Yosida [251, 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.35 (Banach or Closed Range). Let A (V ; W ). The following statements are equivalent: ∈ L Appendices 431

(i) im(A) is closed. (ii) im(A∗) is closed. (iii) im(A) = (ker(A∗))⊥. (iv) im(A∗) = (ker(A))⊥.

Proof. See Brezis [56, Thm. 2.19], Yosida [251, p. 205]. ⊓⊔ We now put in place the second keystone of the ediﬁce: Theorem A.36 (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 [56, Thm. 2.6], Lax [166, p. 168], Rudin [206, p. 47], Yosida [251, p. 75]. ⊓⊔ Theorem A.36, also due to Banach, has far-reaching consequences. In particular, we deduce the following result. Lemma A.37 (Characterization of closed range). Let A (V ; W ). The following statements are equivalent: ∈ L (i) im(A) is closed. (ii) A has a bounded right-inverse map A† : im(A) V , i.e., (A A†)(w)= w → ◦ for all w im(A) and there exists α > 0 such that α A†w w ∈ k kV ≤ k kW for all w im(A) (A† is not necessarily linear). ∈ 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(B (0, 1))→ is open in im(A). Since 0 V ∈ A(BV (0, 1)), there is γ > 0 such that BW (0,γ) A(BV (0, 1)). Let w im(A). Since γ w B (0,γ), there is z B (0, 1)⊂ such that Az = γ ∈w . Set- 2 w W W V 2 w W k k ∈ ∈ k k 2 w W γ ting A†w = k k z leads to A(A†w)= w and A†w w . γ 2 k kV ≤k kW (ii) (i). Let (wn)n N be a sequence in im(A) that converges to some ⇒ ∈ w W . The sequence (vn = A†wn)n N in V is such that Avn = wn and ∈ ∈ α vn V wn W . Then, (vn)n N is a Cauchy sequence in V . Since V is a k k ≤ k k ∈ Banach space, (vn)n N converges to a certain v V . Owing to the bounded- ∈ ∈ ness of A, (Avn)n N converges to Av. Hence, w = Av im(A). ∈ ∈ ⊓⊔ Corollary A.38 (Bounded inverse). If A (V ; W ) is bijective, then 1 ∈ L A− (W ; V ). ∈ L Proof. Since A is bijective, im(A)= W is closed. Moreover, the right-inverse 1 1 A† is necessarily equal to A− (apply A− to A A† = IW ). Lemma A.37(ii) 1 1 ◦ 1 shows that A− (W ; V ) with A− (W ;V ) α− . ∈ L k kL ≤ ⊓⊔ 432 Appendix A. Linear operators in Banach Spaces

A.2.2 Characterization of surjectivity As a consequence of the Closed Range Theorem and of the Open Mapping Theorem, we deduce two characterizations of surjective operators.

Lemma A.39 (Surjectivity of A∗). Let A (V ; W ). The following statements 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 Av α v , v V, (A.16) k kW ≥ k kV ∀ ∈ or, equivalently, there exists α> 0 such that

w′, Av ′ inf sup |h iW ,W | α. (A.17) v V w′ W ′ w′ W ′ v V ≥ ∈ ∈ k k k k Proof. (i) (iii). Since the map A∗ is surjective, Lemma A.37 implies that A∗ ⇒ has a bounded right-inverse map A∗† : V ′ W ′; in particular, A∗(A∗†v′)= v′ → for all v′ V ′, and there is α > 0 such that α A∗†v′ W ′ v′ V ′ . Let now v V . Then,∈ k k ≤ k k ∈

v′, v V ′,V A∗(A∗†v′), v V ′,V 1 A∗†v′, Av W ′ ,W |h i | = |h i | α− |h i | v ′ v ′ ≤ A v ′ k ′kV k ′kV k ∗† ′kW 1 w′, Av W ′,W α− sup |h i |. ≤ w′ W ′ w′ W ′ ∈ k k ′ v ,v V ′,V Taking the supremum in v V yields (A.17) since v = sup ′ ′ |h i | . ′ ′ V v V v′ ′ ∈ k k ∈ V (iii) (ii). The bound (A.16) implies that A is injective. Consider a sequencek k ⇒ (vn)n N such that (Avn)n N is a Cauchy sequence in W . Then, (A.16) implies ∈ ∈ that (vn)n N is a Cauchy sequence in V . 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.35(iv) together with the ⇒ injectivity of A to infer that im(A∗)= 0 ⊥ = V ′. { } ⊓⊔ Lemma A.40 (Surjectivity of A). Let A (V ; W ). The following statements are equivalent: ∈ L (i) A : V W is surjective. → (ii) A∗ : W ′ V ′ is injective and im(A∗) is closed in V ′. (iii) There exists→ α> 0 such that

A∗w′ ′ α w′ ′ , w′ W ′, (A.18) k kV ≥ k kW ∀ ∈ or, equivalently, there exists α> 0 such that

A∗w′, v ′ inf sup |h iV ,V | α. (A.19) ′ ′ w W v V w′ W ′ v V ≥ ∈ ∈ k k k k Appendices 433

Proof. We only detail the proof that (i) (iii); the other two implications are shown as above. Since the map A is surjective,⇒ Lemma A.37 implies that A has a bounded right-inverse map A† : W V ; in particular, A(A†w)= w → for all w W , and there is α> 0 such that α A†w w . Then, for all ∈ k kV ≤k kW w′ W ′, we have ∈ A∗w′, v V ′,V A∗w′, A†w V ′,V A∗w′ V ′ = sup |h i | sup |h i | k k v V v V ≥ w W A†w V ∈ k k ∈ k k w′, w W ′,W w′, w W ′ ,W = sup |h i | α sup |h i | = α w′ W ′ . w W A†w V ≥ w W w W k k ⊓⊔ ∈ k k ∈ k k Remark A.41 (Lions’ Theorem). The statement (i) (iii) in Lemma A.40 is sometimes referred to as Lions’ Theorem. Establishing⇔ the a priori estimate (A.18) 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 . ⊓⊔ Lemma A.42 (Right-inverse). Let V, W be Banach spaces and let A (V ; W ) be a surjective operator. Assume that V is reﬂexive. Then A has∈ a L bounded right-inverse map A† : W V satisfying α A†w V w W , where α is the same constant as in the inf-sup→ condition (A.19)k .k ≤k k Proof. The proof is inspired from ideas by P. Azerad (private communication). Lemma A.34(ii) shows that the adjoint operator A∗ : W ′ V ′ is injective. → Set R := im(A∗) V ′; we equip R with the norm ′ . The injectivity of A∗ ⊂ k·kV implies the existence of a linear left-inverse map A∗‡ : R W ′. Consider its HB → adjoint A∗‡∗ : W ′′ R′. Let E ′ ′′ be one Hahn–Banach extension operator → R V from R′ to V ′′ (see Theorem A.15). Let us set

1 HB A† = J − E ′ ′′ A∗‡∗ J : W V, V ◦ R V ◦ ◦ W → and let us verify that A† satisfies the expected properties; here, we have used that V is reflexive to invoke the inverse of the canonical isometry J : V V → V ′′. We have, for all (w′, w) W ′ W , ∈ × HB w′, AA†w ′ = A∗w′, A†w ′ = E ′ ′′ (A (J (w))), A w ′′ ′ h iW ,W h iV ,V h R V ∗‡∗ W ∗ ′iV ,V = A (J (w)), A w ′ = J (w), A A w ′′ ′ h ∗‡∗ W ∗ ′iR ,R h W ∗‡ ∗ ′iW ,W = J (w), w ′′ ′ = w′, w ′ , h W ′iW ,W h iW ,W where to pass from the first to the second line we have used that A∗w′ R. Moreover, we observe that, for all w W , ∈ ∈ A∗‡∗(JW w), A∗w′ R′,R A†w V = A∗‡∗(JW w) R′ = sup |h i | k k k k w′ W ′ A∗w′ V ′ ∈ k k JW w, w′ W ′′ ,W ′ w′ W ′ = sup |h i | sup k k w W . w′ W ′ A∗w′ V ′ ≤ w′ W ′ A∗w′ V ′ k k ∈ k k ∈ k k 1 We conclude from (A.18) that A†w α− w . k kV ≤ k kW ⊓⊔ 434 Appendix A. Linear operators in Banach Spaces

Remark A.43 (Counter-example). The assumption that V be reflexive in Lemma A.42 cannot be removed, otherwise the exact bound α A†w V p k k ≤ w W is lost. Let us consider the real sequence spaces ℓ , p [1, ]. Since Vk k:= ℓ1 is not reflexive, there exists a linear form A : ℓ1 ∈W :=∞ R that does not attain its norm on the unit ball of V (this is James’s→ Theorem [154, 2 Thm. 1]). Using ℓ as pivot space, it is well-known that ℓ∞ can be identified with the dual of V (see e.g., Brezis [56, Thm. 4.14]). Let t be the nonzero sequence in ℓ∞ such that A(v) = (v,t) 2 := ∞ v t for all v V . A ℓ i=1 i i ∈ simple computation shows that the adjoint A∗ : R V ′ ℓ∞ is such that → ≡ ∗ ′ P (A w ,v)ℓ2 A∗(s) = st for all s R. Let us define α := inf ′ R sup 1 | ′ | ; then w v ℓ w v 1 ∈ ∈ ∈ | |k kℓ (t,v)ℓ2 α := sup 1 | | = A ′ . Let A be a right-inverse of A. Then, for all v ℓ v 1 V † ∈ k kℓ k k A†(s) s R, we have s = (A A†)(s) = (t, A†(s))ℓ2 . For all s R 0 , † ∈ ◦ ∈ \{ } A (s) V is in the unit ball of V . Since A does not attain its norm on thisk ballk by A†(s) A†(s) 1 assumption, we infer that A( † ) < α. Since A( † )= † s, A (s) V A (s) V A (s) V | k k1 | k k k k we can rewrite the above bound as s < A†(s) for all s R 0 . α | | k kV ∈ \{ } ⊓⊔ Let us finally give a sufficient condition for the range of an injective operator to be closed. Lemma A.44 (Peetre–Tartar). Let X, Y , Z be Banach spaces. Let A (X; Y ) be injective and let T (X; Z) be compact. Assume that there is∈ L ∈ L c> 0 such that c x X Ax Y + T x Z for all x X. Then im(A) is closed; equivalently, therek isk α>≤k0 suchk thatk k ∈

α x Ax , x X. (A.20) k kX ≤k kY ∀ ∈ Proof. Owing to Lemma A.39 and since A is injective, im(A) is closed iﬀ (A.20) holds. We prove (A.20) by contradiction. Assume that there is a sequence (xn)n N of X such that xn X = 1 and Axn Y converges to zero ∈ k k k k when n goes to inﬁnity. Since T is compact and the sequence (xn)n N is ∈ bounded, there is a subsequence (xnk )k N such that (T xnk )k N is a Cauchy sequence in Z. Owing to the inequality ∈ ∈

α x x Ax Ax + T x T x , 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.2.3 Characterization of bijectivity The following theorem provides the theoretical foundation of the BNB Theo- rem of 21.3. § Theorem A.45 (Bijectivity of A). Let A (V ; W ). The following statements are equivalent: ∈ L Appendices 435

(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.21) k kW ≥ k kV ∀ ∈

or, equivalently, A∗ is injective and

w′, Av ′ inf sup |h iW ,W | =: α> 0. (A.22) v V w′ W ′ w′ W ′ v V ∈ ∈ k k k k Proof. (1) Statements (ii) and (iii) are equivalent since (A.21) is equivalent to A injective and im(A) closed owing to Lemma A.39. (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. ⊓⊔ 1 Remark A.46 (Bounds on A− ). Let A (V ; W ) be a bijective operator. 1 ∈ L Then A− (W ; V ) owing to Corollary A.38. Moreover, using the bijectivity of A, we have∈ L

1 1 1 A− w − v − sup k kV = sup k kV w W w W v V Av W ∈ k k ∈ k k Av w′, Av ′ = inf k kW = inf sup |h iW ,W |, v V v V v V w′ W ′ w′ W ′ v V ∈ k k ∈ ∈ k k k k 1 1 which shows using (A.22) that A− (W ;V ) = α− . Similarly, we have k kL 1 1 A− w V − Av W inf k k = sup k k = A (V ;W ), w W w W v V v V k kL ∈ k k ∈ k k 1 1 which shows that A− w A − w for all w W (com- k kV ≥ k k (V ;W )k kW ∈ pare with (A.21)) and that the followingL inf-sup condition holds (compare with (A.22)): 1 v′, A− w V ′,V 1 inf sup |h i | = A −(V ;W ). w W v′ V ′ v′ V ′ w W k kL ⊓⊔ ∈ ∈ k k k k

Remark A.47 (Bijectivity of A∗). The bijectivity of A (V ; W ) is ∈ L equivalent to that of A∗ (W ′; V ′). Indeed, statement (ii) in Theorem A.45 ∈ L is equivalent to A∗ injective and A∗ surjective owing to the equivalence of statements (i) and (ii) from Lemma A.39. ⊓⊔ 436 Appendix A. Linear operators in Banach Spaces

Corollary A.48 (Inf-sup condition). Let A (V ; W ) be a bijective operator. Assume that V is reﬂexive. Then, ∈ L

w′, Av ′ w′, Av ′ inf sup |h iW ,W | = inf sup |h iW ,W |. (A.23) ′ ′ v V w′ W ′ w′ W ′ v V w W v V w′ W ′ v V ∈ ∈ k k k k ∈ ∈ k k k k Proof. The left-hand side, l, and the right-hand side, r, of (A.23) are two positive 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 l v for all v in V . Let w′ W ′ and w W . Since A is surjective, k kW ≥ k kV ∈ ∈ we can consider its right-inverse map A†, and the previous statement regarding l implies that l A†w w . Since A(A†w)= w, this implies that k kV ≤k kW

w′, w W ′ ,W A∗w′, A†w V ′,V w′ W ′ = sup |h i | = sup |h i | k k w W w W w W w W ∈ k k ∈ k k A†w V 1 A∗w′ V ′ sup k k A∗w′ V ′ , ≤k k w W w W ≤ l k k ∈ k k so that l r. The other inequality r l is proved similarly by working with ≤ ≤ W ′ in lieu of V , V ′ in lieu of W and A∗ in lieu of A, leading 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 by using the reﬂexivity of V . ⊓⊔ Remark A.49 (Counter-example). (A.23) may not hold if A = 0 is not 6 bijective. For instance if A : (x1, x2, x3 ...) (0, x1, x2, x3,...) is the right 2 7→ shift operator in ℓ , then A∗ : (x , x , x ...) (x , x , x ,...) is the left shift 1 2 3 7→ 2 3 4 operator. It can be veriﬁed that A is injective but not surjective whereas A∗ is surjective but not injective. It can also be shown that l = 1 and r = 0. ⊓⊔ A.2.4 Coercive operators We now focus on the smaller class of coercive operators.

Deﬁnition A.50 (Coercive operator). Let V be a complex Banach space. The operator A (V ; V ′) is said to be a coercive if there exist a real number α> 0 and a complex∈ L number ξ with ξ =1 such that | | 2 (ξ Av, v ′ ) α v , v V. (A.24) ℜ h iV ,V ≥ k kV ∀ ∈ 2 In the real case, Av, v ′ α v for all v V . ±h iV ,V ≥ k kV ∈ Remark A.51 (Modulus). The coercivity condition is sometimes written 2 as Av, v ′ α v for all v V . While this variant looks slightly more |h iV ,V |≥ k kV ∈ general since (ξ Av, v ′ ) Av, v ′ , it turns out to be equivalent ℜ h iV ,V ≤ |h iV ,V | Appendices 437 to (A.24). Let us prove the claim in the real case. It suﬃces to show that 2 Av, v V ′,V α v V implies that Av, v V ′,V has always the same sign for all nonzero|h i v |≥V . Reasoningk k by contradiction,h i if there are nonzero v, w V such ∈ ∈ that Av, v ′ < 0 and Aw, w ′ > 0, then the second-order polynomial h iV ,V h iV ,V R λ A(v + λw), v + λw V ′,V R has at least one root λ R; coercivity with∋ absolute7→ h value yields v +i λ w∈= 0 so that a(v, v)= λ2a(w,∗ ∈ w) > 0, which contradicts a(v, v) < 0. For the∗ complex case, see Brezis [56,∗ p. 366]. ⊓⊔ The following proposition shows that the notion of coercivity is relevant only in Hilbert spaces. Proposition A.52 (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. Setting ((v, w))V := ξ Av, w V ′,V + ξ Aw, v V ′,V defines a sesquilinear map on V V with Hermitianh symmetry.i Theh coercivityi and boundedness of A imply that× 2 2 2α v V ((v, v))V 2 A (V ;V ′) v V , k k ≤ ≤ k kL k k for all v V . This shows positive-definiteness (so that (( , ))V is an inner product in∈V ) and that the induced norm is equivalent to · · . k·kV ⊓⊔ Corollary A.53 (Sufficient condition). Coercivity is a sufficient condition for an operator A (V ; V ′) to be bijective. ∈ L Proof. Corollary A.53 is the Lax–Milgram Lemma; see 21.2. § ⊓⊔ Corollary A.54 (Self-adjoint bijective operator). Assume that V is reflexive. Let A (V ; V ′) be a self-adjoint operator. Then, A is bijective if and only if there∈ isL a real number α> 0 such that

Av ′ α v , v V. (A.25) k kV ≥ k kV ∀ ∈ Proof. Owing to Theorem A.45, the bijectivity of A implies that A satisfies inequality (A.25). Conversely, inequality (A.25) means that A is injective. 1 It follows that A∗ is injective since A∗ = A JV− owing to the reflexivity hypothesis. The conclusion is then a consequence◦ of Theorem A.45(iii). ⊓⊔ Definition A.55 (Monotone operator). The operator A (V ; V ′) is said to be monotone if ∈ L

( Av, v ′ ) 0, v V. (A.26) ℜ h iV ,V ≥ ∀ ∈ Corollary A.56 (Equivalent condition). Assume that V is reﬂexive. 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 21.5. ⊓⊔ 438 Appendix A. Linear operators in Banach Spaces A.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, e.g., Bergh and Löfström [22], Tartar [232] and references therein. For sim- plicity, we focus here on the real interpolation K-method; see [22, 3.1] and [232, Chap. 22]. § Let V0 and V1 be two normed vector spaces, continuously embedded into a common topological vector space . 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 [22, 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.27) 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.57 (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.28) { ∈ |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.58 (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.59 (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 [22, p. 39]. 7→ ⊓⊔ Remark A.60 (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 ⊓⊔