Some of you have been querying me about the proof of Lemma 4.1.1. I wrote in the guidance for the second test that it is important to study this proof. Of course, there will never be a question on the test (or exam) like "prove lemma 4.1.1". First of all it is too long, but secondly I do not believe in remembering such proofs by heart. However, the manipulations and arguments used are very instructive of similar but even more involved estimates of this kind one encounters in the analysis of ordinary and partial differential equations in various contexts, so although I do not want you to learn this proof by heart, I would really appreciate if you understand it when you read it.... I posted a video in the left-hand side margin where I go a bit slower through the proof than in the lecture notes or in the lecture, and I hope it helps you if you have been struggling with some of the steps in the proof.

## Monday, 7 March 2016

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