Discussion Board Lecture 13
Two misprints
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Dear all, 1. on p. 151, line beginning with `Showing that there ...' the inequality should be $ \|Bf\|_2\le c\|f\|_2 $. 2. Exercise 13.2(a) should end in `... is dense in $ L_2^0(\Omega) $'. Best wishes, Jürgen
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Dear Jürgen, I guess there is also a misprint in Theorem 13.6 and in the line (same page) beginning with 'In this form': $ Bf\in C_c^{\infty}(\mathbb{R}^n;\mathbb{K}^n) $. Best, Jamil
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Dear Jamil, thanks for the communication! You are absolutely correct. (I apologise: it is easy to get confused with the vector valued function spaces.) And here are two additional misprints: The last line of the proof of Theorem 13.12 should be: $ \mathop{\rm ran}(A^*)=\mathop{\rm ran}(A_1^*) $ ... On the first line of p. 154 one should have ... $ B(y',r) $. Best wishes, Jürgen
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Second solution to Exercice 13.2(b)
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Dear Sydney team, this is to mention that there is a flaw in your second solution to Exercise 13.2(b): the problem is that the sum of closed subspaces of a Hilbert space need not be closed. Here is a counterexample: Let $ H:=\ell_2(\mathbb Z\setminus\{0\}) $, $ H_1 $ the closed linear span of $ \{e_n;\,n\in\mathbb N\} $ and $ H_2 $ the closed linear span of $ \{e_n+\frac1ne_{-n};\,n\in\mathbb N\} $ (where $ e_n $ are the standard unit vectors. Then clearly $ H_1+H_2 $ is dense in $ H $. But trying to find a decomposion of $ \sum_{n\in\mathbb N}\frac1ne_{-n} $ as a sum of two elements in $ H_1 $ and $ H_2 $ is not successful. On a higher level: If $ H_1+H_2 $ would be closed, then it would be equal to $ H $, and $ H $ would be the topological direct sum of $ H_1 $ and $ H_2 $. However, for $ n\in\mathbb N $ the decomposition of $ e_{-n} $ in $ H_1+H_2 $ is given by $ e_{-n}=-ne_n+(ne_n+e_{-n}) $, and this formula shows that the projections are not continuous. Best wishes, Jürgen
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