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Q: Equality of norms $\|\sum_j a_j u_j \|\leq \sum_j \|a_j u_j\|$ If $u_j$ are unit vectors in an infinite dimensional Hilbert space, how to prove the following? $$\|\sum_j a_j u_j \|\leq \sum_j \|a_j u_j\|$$ I guess that we may need Schur's lemma, but it seems the only way that I can see is to use the fact that they are normalized. A: If the $u_j$ form a Bessel sequence with bound $B>0$, then $$\|\sum a_ju_j\|^2\le \sum \|a_j\|^2\|u_j\|^2 \le \sum\|a_j\|^2B^2=\|\sum a_j\|^2B^2.$$ This shows that the $u_j$ are almost orthonormal. From the last inequality we can now show that $\|\sum a_ju_j\|^2\le \sum \|a_j\|^2\|u_j\|^2$. Now let $\|\sum a_ju_j\|=\alpha$ and $\|\sum a_ju_j\|^2\le \sum \|a_j\|^2\|u_j\|^2$. Then $\alpha^2\le \sum\|a_j\|^2B^2$, so $\alpha\le \sqrt{\sum\|a_j\|^2B^2}$. Q: Python Multiprocessing Pool.map with keys The purpose of this script is to return all the words that have the same first letter in a given list. What I want is to be able to use the wordlist and the function within the function Pool.map(
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Windows 7 Windows 8 Windows 8.1 Mac OS X 10.6 or later Linux 2.6 or later ***Please take your time to read the list of content provided, and try to find the bugs we’ve found so far. Disclaimer: Please note that this is the first beta release and all errors and bugs may be present in this build. We are always grateful for your feedback and ideas to make the game better! Please do not report bugs you encounter in the game directly
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