Isem-comment-Nov09-1-daniel

1Dear ISEM team,We have a few comments regarding lectures one and two. It is ouropinion that Lemmas 1.2, 2.1, and 2.2 could be presented in a more con-structive way. We therefore propose the following formulation which (1)gives more information about the transformation Q mentioned in Lemmad d 02.2 of lecture two and (2) shows how the spaceR and its dual space (R )are isomorphic to each other using the expression for Q and the transfor-mation J as given in Lemma 1.2 of lecture one and Lemma 2.1 of lecturetwo respectively. One of the motivations for this formulation is to introducea numerical aspect among the topics in this course. As is known, the roleof gradients plays an important part in numerical analysis and in particu-lar optimization problems (see for example the work John W. Neuberger,chapter 8 in Sobolev Gradients and Differential Equations, Lecture Notesin Mathematics 1670, Springer-Verlag (1997)). We thus suggest an exercisethat was taken from a course given by John Neuberger to demonstrate therelationship between gradient systems in an infinite dimensional Hilbertspace setting and finite dimensional spaces.We first propose the following Lemma 2.2 , which gives you more in-formation about the operator Q mentioned in the original Lemma 2.2 oflecture two: dLemma 2.2 For every inner producth ; i onR there exists an iso-d dmorphism Q :R ! R having the following propertiesd(i)hv;wi =hQv;wi for all v;w2R ,euc(ii) Q is symmetric,(iii) Q is ...