ABSTRACT
This introductory chapter gives some known facts that will be needed later. The reader can skip it on the first reading and return to it only after being prompted by a reference.
Lemma 1.1. 1◦ A linear bounded operator A acting from one Banach space X into another Y is boundedly invertible if and only if
l(A) := min ‖x‖=1
‖Ax‖ > 0 . (1.1)
In this case,
∥∥A−1∥∥ = 1 l(A)
. (1.2)
2◦ For every two linear operators A and B ,∣∣l(A)− l(B)∣∣ ≤ ‖A−B‖ . Proof. 1◦ Let A be boundedly invertible. If l(A) = 0, then there exists a sequence xn with ‖xn‖ = 1 & ‖Axn‖ < 1/n , so that
∥∥A−1∥∥ := sup ‖y‖=1
∥∥A−1y∥∥ ≥ ∥∥∥∥A−1 Axn‖Axn‖ ∥∥∥∥ = 1‖Axn‖ > n .
