Discussion Board Lecture 09
Density needed on remark 9.15
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Dear all, In remark 9.15, I think you need, more than the continuos embedding, density of V in H for justifying why the weak convergence in V imply weak convergence in H. In fact, in section 9.4 there where we apply this remark we have this density (a is j-elliptic and j have dense range). Best wishes, Abdallah.
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Dear Abdallah, no denseness is needed. If $ (u_n) $ is weakly convergent to $ u $ in $ V $, then $ \eta(u_n)\to\eta(u) $ for all continuous linear functionals $ \eta $ on $ V $. If $ v\in H $, then $ V\ni w\mapsto(w\mid v) $ is a continuous linear functional on $ V $, and therefore $ (u_n\mid v)\to(u\mid v) $. The latter, for all $ v\in H $, is just the weak convergence of $ (u_n) $ to $ u $ in $ H $. Best wishes, Jürgen
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(9.7)
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Dear all, correction: At the end of (9.7) one should read `($ u\in H,\ v\in V $)'. In this form it is needed in the next displayed formula. Best wishes, Jürgen
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Misprints in Remark 9.10
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Dear all, here is a correction of misprints on line 4 of Remark 9.10: ... ($ t\geqslant2/k $), $ F_k(t)=-1/k $ ($ t\leqslant-2/k $), we ... and on line 5: ... locally in $ L_1 $ on $ (a,b) $. ...
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Reference in Example 9.21
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Dear participants, due to a mistake (multiply defined label) we have a reference to Example 9.21 in Example 9.21. The reference should be to Example 9.8. Sorry! Best wishes, Jürgen
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Dear participants, we have uploaded a corrected version of Lecture 9. Best wishes, Christian
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