A Peetre–Tartar Equivalence Theorem
Thierry Goudon · 2016
The purpose of this appendix is to study the following lemma and some of its applications.LEMMA A3.1.-LetE be a Banach space and let F, G be normed spaces.Let A ∈ L (E, F ) and B ∈ L (E, G).We assume that: i) B is compact; ii) There exist μ, ν > 0, such that for all x ∈ E, we haveThus, Ker(A) is of finite dimension and Ran(A) is closed.PROOF.-The inequality on the left-hand side in ii) simply expresses the continuity of the operators A and B. The inequality on the right-hand side proves thatWe deduce from this that B Ker(A) is injective and thus 0 is not an eigenvalue of B Ker(A) .Let u n n∈N be a normed sequence of elements of Ker(A): for all n ∈ N, we have Au n = 0 andWe have thus shown that the unit sphere of Ker(A) is compact, which implies, according to Riesz's lemma (see [GOU 11, Theorem 5.13]), that Ker(A) is of finite dimension.