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Appendix a Banach and Hilbert Spaces A.1 Banach Spaces Appendix A Banach and Hilbert spaces The goal of this appendix is to recall basic results on Banach and Hilbert (JLG) Sept 15 2019 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 recov- ered by replacing the field C by R, by removing the real part symbol ( ) and the complex conjugate symbol , and by interpreting the symbol ℜ as· the absolute value instead of the modulus.· |·| A.1 Banach spaces Let V be a complex vector space. Definition A.1 (Norm). A norm on V is a map V : V R+ := [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 ∈ For every norm V : V R+ := [0, ), the function d(x, y) := x y V , for all x, y V ,k·k is a metric→ (or distance).∞ k − k ∈ Remark A.2 (Definiteness). Item (i) can be slightly relaxed by requiring only that [ v V = 0] = [ v = 0 ], since the 1-homogeneity implies that [ v =0]= k[ kv = 0 ]. ⇒ ⇒ k kV ⊓⊔ Definition A.3 (Seminorm). A seminorm on V is a map V : V R+ satisfying only the statements (ii) and (iii) above, i.e., 1-homogeneity|·| and→ the triangle inequality Definition A.4 (Banach space). A vector space V equipped with a norm is called Banach space if every Cauchy sequence in V has a limit in V . k·kV 378 Appendix A. Banach and Hilbert spaces Definition A.5 (Equivalent norms). Two norms V,1 and V,2 are said to be equivalent on V if there exists a positive realk·k number c suchk·k that 1 c v v c− v , v V. (A.1) k kV,2 ≤k kV,1 ≤ k kV,2 ∀ ∈ Whenever (A.1) holds true, V is a Banach space for the norm V,1 if and only if it is a Banach space for the norm . k·k k·kV,2 Remark A.6 (Finite dimension). If 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 [46, Thm. 6.5], Lax [126, 5.2]. § ⊓⊔ A.2 Bounded linear maps and duality Definition A.7 (Linear, antilinear map). 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 v , v V and→ A(λv) = λA(v) for all λ C and all v V , and 1 2 ∈ ∈ ∈ it is said to be antilinear if A(v1 + v2) = A(v1)+ A(v2) for all v1, v2 V and A(λv) = λAv for all λ C and all v V . To simplify the notation,∈ we sometimes write Av instead∈ of A(v). ∈ Definition A.8 (Bounded (anti)linear map). Assume that V and W are equipped with norms V and W , respectively. The (anti)linear map A : V W is said to be k·kbounded ork·kcontinuous if → A(v) W A (V ;W ) := sup k k < . (A.2) k kL v V v V ∞ ∈ k k In this book we systematicaly abuse the notation by implicitly assuming that the argument in this type of supremum is nonzero. Bounded (anti)linear maps in Banach spaces are called operators. The complex vector space composed of the bounded linear maps from V to W is denoted (V ; W ). One readily verifies that the map (V ;W ) defined in (A.2) is indeedL a norm on (V ; W ). k·kL L Proposition A.9 (Banach space). Assume that W is a Banach space. Then (V ; W ) equipped with the norm (A.2) is also a Banach space. The same statementL holds true for the complex vector space composed of all the bounded antilinear maps from V to W . Proof. See Rudin [162, p. 87], Yosida [193, p. 111]. ⊓⊔ Example A.10 (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 . This means that the embedding of V into W is continuous.k k ≤ k Wek say that V∈ is continuously . embedded into W , and we write V ֒ W → ⊓⊔ Appendices 379 The dual of a real Banach space V is composed of the bounded linear maps from V to R. The same definition can be adopted if V is a complex space, but to stay consistent with the formalism considered in the weak formulation of complex-valued PDEs, we define the dual space as being composed of bounded antilinear maps from V to C. Definition A.11 (Dual space). Let V be a complex vector space. The dual space of V is denoted V ′ and is composed of the bounded antilinear maps from V to C. An element A V ′ is called bounded antilinear form, and its ∈ action on an element v V is denoted either A(v) or A, v ′ . ∈ h iV ,V Owing to Proposition A.9, V ′ is a Banach space with the norm A(v) A, v V ′,V A V ′ = sup | | = sup |h i |, A V ′. (A.3) k k v V v V v V v V ∀ ∈ ∈ k k ∈ k k Remark A.12 (Linear vs. antilinear form). If A : V C is an antilinear form, then A (defined by A(v) := A(v) C for all v V )→ is a linear form. ∈ ∈ ⊓⊔ A.3 Hilbert spaces Let V be a complex vector space. Definition A.13 (Inner product). An inner product on V is a map ( , )V : V V C satisfying the following three properties: (i) Sesquilin- earity· · (the× prefix→ sesqui means one and a half): ( , w) is a linear map for · V all fixed w V , whereas (v, )V is an antilinear map for all fixed v V . If V is a real∈ vector space, 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, v)V = 0 ] [ v = 0 ]. (Notice that (v, v)V is always real owing to the Hermitian symmetry⇐⇒ and that (0, ) = ( , 0) =0 owing to sesquilinearity.) · V · V Proposition A.14 (Cauchy–Schwarz). Let ( , )V be an inner product on V . By setting · · 1 v := (v, v) 2 , v V, (A.4) k kV V ∀ ∈ one defines a norm on V . This norm is said to be induced by the inner product. Moreover we have the Cauchy–Schwarz (v, w) v w , v, w V. (A.5) | V |≤k kV k kV ∀ ∈ Definition A.15 (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). 380 Appendix A. Banach and Hilbert spaces Theorem A.16 (Riesz–Fr´echet). Let V be a complex Hilbert space. For all A V ′, there exists a unique v V s.t. (v, w) = A, w ′ for all ∈ ∈ V h iV ,V w V , and we have v = A ′ . ∈ k kV k kV Proof. See Brezis [46, Thm. 5.5], Lax [126, p. 56], Yosida [193, p. 90]. ⊓⊔ A.4 Compact operators Definition A.17 (Compact operator). Let V, W be two complex Banach spaces. The operator T (V ; W ) is said to be compact if from every ∈ L bounded sequence (vn)n N in V , one can extract a subsequence (vn )k N such ∈ k ∈ that the sequence (T (vnk ))k N converges in W . Equivalently, T maps the unit ball in V into a relatively compact∈ set in W (that is, a set whose closure is compact). Example A.18 (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.19 (Composition). Let W , X, Y , Z be four Banach spaces and let A (Z; Y ), K (Y ; X), B (X; W ) be three operators. Assume that K is∈ compact. L Then∈ the L operator ∈B L K A is compact. ◦ ◦ The following compactness result is used at several instances in this book. The reader is referred to Tartar [181, Lem. 11.1] and Girault and Raviart [103, Thm. 2.1, p. 18] for a slightly more general statement and references. Lemma A.20 (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. Assume that there is c> 0 such that c x X A(x) Y + T (x) Z for all x X. Then there is α> 0 such that k k ≤k k k k ∈ α x A(x) , x X. (A.6) k kX ≤k kY ∀ ∈ Proof. We prove (A.6) by contradiction. Assume that there is a sequence (xn)n N of X s.t. xn X = 1 and A(xn) Y converges to zero as n . Since ∈ k k k k → ∞ T is compact and the sequence (xn)n N is bounded, there is a subsequence ∈ (xn )k N s.t.
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