Dirk
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 1d comment Closed subset for $L^2$ strong and weak convergence Yep, you are in the right track. 1d comment Closed subset for $L^2$ strong and weak convergence Do you know the weak limit of $f_n(x) = \sin(nx)$? If so, try to find a similar sequence with $f_n(x) \in K=\{-1,1\}$. Feb 4 comment Justify or provide counterexamples My point is, that the function is equal to $e^{-x}$ (almost everywhere which is true equality in the space $L^1$). Hence, it is not an example since $e^{-x}\to 0$. Feb 4 answered Justify or provide counterexamples Feb 4 comment Justify or provide counterexamples Strictly speaking your example in (i) is not a counterexample, as it is the same function as $e^{-x}$ (in $L^1$). Feb 4 comment Mentally approximating an inverse sine? I don't know what unit you use. It's $\pi/2$ for radiants, $180$ for degree. Feb 4 comment Mentally approximating an inverse sine? $\sin(x) = \cos(x + c)$ ($c$ depending if radiants, degree or whatever). Hence, for the inverse (on appropriate domains): $\sin^{-1}(x) = \cos^{-1}(x)-c$. Jan 27 comment What is wrong with my contradiction? Something to memorize: Induction shows the result for an arbitrarily large number of summands. But "arbitrarily large" is still not quite infinity. Jan 20 revised why don't standard analysis texts relax the injectivity requirement of the multivariable change of variables theorem? added 22 characters in body Jan 2 answered why don't standard analysis texts relax the injectivity requirement of the multivariable change of variables theorem? Dec 20 reviewed Close can't understand proof for the law of tangents Dec 20 reviewed Leave Open Show that $\lim\limits_{r\to\infty} \textrm{\{} \|x-ru\|-\|y-ru\| \textrm{}\} = \left$ Dec 20 reviewed Close Example of a series, which its square sum converges, but it is not a Cauchy sequence. Dec 15 awarded Explainer Dec 15 revised Derivative of 1-norm edited title Dec 15 answered Derivative of 1-norm Dec 14 reviewed Leave Open Is there a surjective group homomorphism $GL_{n}(k) \to GL_{m}(k)$ where $n > m$? Dec 14 reviewed Close The relation between face counts and edge counts in a polyhedron, $3f_3 + 4f_4 + 5f_5 +\dots = 2E$ Dec 9 comment What comes after geometric mean? Other problems besides non commutativity are that the $n$-th super root does not always exists (even for positive arguments) and that it is in general multivalued (and even two different positive super roots can occur). Dec 9 answered Taking the means of numbers repeatedly