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Shapes and Diffeomorphisms, Applied Mathematical Sciences 171, 424 Appendix A: Elements from Functional Analysis

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Shapes and Diffeomorphisms, Applied Mathematical Sciences 171, 424 Appendix A: Elements from Functional Analysis Appendix A Elements from Functional Analysis This first chapter of the appendix includes results in functional analysis that are needed in the main part of the book, focusing mostly on Hilbert spaces, and providing proofs when these are simple enough. Many important results of the theory are left aside, and the reader is referred to the many treatises on the subject, including [45, 74, 246, 306] and other references listed in this chapter, for a comprehensive account. A.1 Definitions and Notation Definition A.1 AsetH is a (real) Hilbert space if: (i) H is a vector space on R. ( , ) → , , ∈ (ii) H has an inner product denoted h h h h H ,forh h H. This inner product is a symmetric positive definite bilinear form. The associated norm is = , denoted h H h h H . (iii) H is a complete space with respect to the topology associated to the norm. · If condition (ii) is weakened to the fact that H is a norm (not necessarily induced by an inner product), one says that H is a Banach space. Convergent sequences in the norm topology are sequences hn for which there ∈ − → exists an h H such that h hn H 0. Property (iii) means that if a sequence (hn, n ≥ 0) in H is a Cauchy sequence, i.e., it collapses in the sense that, for every > − ≤ ≥ positive ε there exists an n0 0 such that hn hn0 H ε for n n0, then it ∈ − → necessarily has a limit: there exists an h H such that hn h H 0asn tends to infinity. If H satisfies (i) and (ii), it is called a pre-Hilbert space. On pre-Hilbert spaces, the Schwarz inequality holds: Proposition A.2 (Schwarz inequality) If H is pre-Hilbert, and h, h ∈ H, then , ≤ . h h H h H h H © Springer-Verlag GmbH Germany, part of Springer Nature 2019 423 L. Younes, Shapes and Diffeomorphisms, Applied Mathematical Sciences 171, https://doi.org/10.1007/978-3-662-58496-5 424 Appendix A: Elements from Functional Analysis The first consequence of this property is: Proposition A.3 The inner product on H is continuous in the norm topology. × → R ( , ) = , Proof The inner product is a function H H . Letting ϕ h h h h H ,we have, by the Schwarz inequality, and introducing a sequence (hn) which converges to h ( , ) − ( , ) ≤ − → , ϕ h h ϕ hn h h hn H h H 0 which proves the continuity with respect to the first coordinate, and also with respect to the second coordinate by symmetry. Working with a complete normed vector space is essential when dealing with infinite sums of elements: if (hn) is a sequence in H, and if hn+1 +···+hn+k can be made arbitrarily small for large n and any k > 0, then completeness implies that ∞ the series n=0 hn has a limit in h. In particular, absolutely converging series in H converge: < ∞⇒ hn H hn converges. (A.1) n≥0 n≥0 (In fact, (A.1) is equivalent to (iii) in normed spaces.) We add a fourth condition to Definition A.1. (iv) H is separable in the norm topology: there exists a countable subset S inH such ∈ > ∈ S − ≤ that, for any h H and any ε 0, there exists an h such that h h H ε. In the following, Hilbert spaces will always be separable without further mention. An isometry between two Hilbert spaces H and H is an invertible linear map : → , ∈ ( ), ( ) = , ϕ H H such that, for all h h H, ϕ h ϕ h H h h H . A Hilbert subspace of H is a subspace H of H (i.e., a non-empty subset, invariant under linear combination) which is closed in the norm topology. Closedness implies that H is itself a Hilbert space, because Cauchy sequences in H are also Cauchy sequences in H, hence converge in H, hence in H because H is closed. The next proposition shows that every finite-dimensional subspace is a Hilbert subspace. Proposition A.4 If K is a finite-dimensional subspace of H, then K is closed in H. Proof Let e1,...,ep be a basis of K .Let(hn) be a sequence in K which converges ∈ ∈ ( ) to someh H; we need to show that h K . We have, for some coefficients akn , = p = ,..., hn k=1 aknek and for all l 1 p: p , = , . hn el H ek el H akn k=1 p p Let an be the vector in R with coordinates (akn, k = 1,...,p) and un ∈ R with ( , , = ,..., ) coordinates hn ek H k 1 p . Let also S be the matrix with coefficients Appendix A: Elements from Functional Analysis 425 = , = skl ek el H , so that the previous system may be written: un San. The matrix S is invertible: if b belongs to the null space of S, a quick computation shows that p 2 T b Sb = bi ei = 0, H i=1 which is only possible when b = 0, because (e1,...,en) is a basis of K . We therefore = −1 ( , , = ,..., ) have an S un. Since un converges to u with coordinates h ek H k 1 p = −1 (by continuity of the inner product), we obtain the fact that an converges to a S u. p → p But this implies that k=1 aknek k=1 ak ek and, since the limit is unique, we have = p ∈ h k=1 ak ek K . A.2 First Examples A.2.1 Finite-Dimensional Euclidean Spaces = Rn , = T = n H with h h Rn h h i=1 hi hi is the standard example of a finite- dimensional Hilbert space. A.2.2 The 2 Space of Real Sequences = ( , ,...) ∞ 2 < ∞ Let H be the set of real sequences h h1 h2 such that i=1 hi . Then H is a Hilbert space, with dot product ∞ , = . h h 2 hi hi i=1 A.2.3 The L2 Space of Functions Let k and d be two integers. Let Ω be an open subset of Rk . We define L2(Ω, Rd ) as the set of all square integrable functions h : Ω → Rd , with inner product , = ( ) ( ) . h h 2 h x h x dx Ω Integrals are taken with respect to the Lebesgue measure on Ω, and two functions which coincide everywhere except on a set of null Lebesgue measure are identified. The fact that L2(Ω, Rd ) is a Hilbert space is a standard result in integration theory. 426 Appendix A: Elements from Functional Analysis A.3 Orthogonal Spaces and Projection Let O be a subset of H. The orthogonal space to O is defined by ⊥ = ∈ :∀ ∈ , , = . O h H o O h o H 0 Theorem A.5 O⊥ is a Hilbert subspace of H. Proof Stability under linear combination is obvious, and closedness is a consequence of the continuity of the inner product. When K is a subspace of H and h ∈ H, one can define the variational problem: (P ( )) : ∈ − = − : ∈ . K h find k K such that k h H min h k H k K The following theorem is fundamental: Theorem A.6 If K is a closed subspace of H and h ∈ H, (PK (h)) has a unique solution, k, characterized by the conditions: k ∈ K and h − k ∈ K ⊥. Definition A.7 The solution of problem (PK (h)) in the previous theorem is called the orthogonal projection of h on K and denoted πK (h). Proposition A.8 πK : H → K is a linear, continuous transformation and πK ⊥ = id − πK . Proof We prove the theorem and the proposition together. Let = − : ∈ . d inf h k H k K If k, k ∈ K , a direct computation shows that k + k 2 − + − 2 = − 2 + − 2 . h k k H 4 h k H h k H 2 2 H 2 The fact that (k + k)/2 ∈ K implies h − k+k ≥ d, so that 2 H 1 k − k2 ≤ h − k2 + h − k2 − d2 . H 2 H H Now, from the definition of the infimum, one can find a sequence kn in K such that − 2 ≤ 2 + −n kn h H d 2 for each n. The previous inequality implies that − 2 ≤ −n + −m , kn km H 2 2 2 Appendix A: Elements from Functional Analysis 427 which implies that kn is a Cauchy sequence, which therefore converges to a limit − = k that belongs to K (because K is closed) and is such that k h H d.Ifthe minimum is attained for another k in K , we have, by the same inequality 1 k − k2 ≤ d2 + d2 − d2 = 0, H 2 = so that k k , uniqueness is proved, hence πK is well-defined. = ( ) ∈ ( ) = − − 2 Let k πK h , k K and consider the function f t h k tk H , which = ( ) = − 2 − is by construction minimal for t 0. We have f t h k H − , + 2 2 − , = 2t h k k H t k H and this can be minimal at 0 only if h k k H 0. Since this has to be true for every k ∈ K , we obtain the fact that h − k ∈ K ⊥. Conversely, if k ∈ K and h − k ∈ K ⊥, we have for any k ∈ K − 2 = − 2 + − 2 ≥ − 2 , h k H h k H k k H h k H so that k = πK (h). This proves Theorem A.6. We now prove Proposition A.8:leth, h ∈ H and α, α ∈ R.Letk = πK (h), k = πK (h ); we want to show that πK (αh + α h ) = αk + α k , for which it suffices to prove (since αk + αk ∈ K ) that αh + αh − αk − αk ∈ K ⊥.
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