AppendixReminders
We give here the precise statements of the definitions1 and results from Volumes 1, Banach, Fréchet, Hilbert and Neumann spaces [81], and 2, Continuous functions [82], in the series Analysis for PDEs that are used in this book, in order to keep it self-contained. Proofs are provided in the said volumes, to which we refer by [Vol. 1, Theorem 12.2] for example. Some statements are limited to simpler cases when that is all we will need, whereas others are complete and proven.
A.1. Notation and numbering
Set theory notation. We use the standard notation (it is presented in § 1.1 of Volume 1). We only recall the difference in notation between the set {a, b, . . . , z} and the ordered set (a, b, . . . , z). Thus, (a, b) ≠ (b, a) if a ≠ b whereas we always have {a, b} = {b, a}.
Countability. We assume that the sets 
of natural numbers2 and 
of integers, respectively
and the set 
of rational numbers are familiar, as well as their addition, subtraction, multiplication, absolute values and orders. We denote and
A sequence in a set U is the data, for every n ∈ , of an element ...
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