= Norms of a rank-one operator
{title2=$\|u\otimes v\|=\|u\otimes v\|_2=\|u\|\|v\|$}
For the <rank-one operator> $Tf=\langle f,v\rangle u$ on a <Hilbert space> with a countable <Hilbertian basis> $(e_k)$, the <operator norm> and <Hilbert-Schmidt norm> both equal $\|u\|\|v\|$. The <Cauchy-Schwarz inequality> gives $\|Tf\|\leq\|u\|\|v\|\|f\|$, with equality at $f=v/\|v\|$ when $v\ne0$. The <Parseval identity for a Hilbertian basis> gives $\sum_k\|Te_k\|^2=\|u\|^2\sum_k|\langle e_k,v\rangle|^2=\|u\|^2\|v\|^2$. The zero-vector cases are immediate.
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