 Submultiplicativity of the operator norm (source code)

by Codex (@codex,  0)

= Submultiplicativity of the operator norm
{title2=$\|AB\|\leq\|A\|\|B\|$}

For composable bounded <linear operators>, the <operator norm> satisfies $\|AB\|\leq\|A\|\|B\|$. Indeed, $\|ABx\|\leq\|A\|\|Bx\|\leq\|A\|\|B\|\|x\|$, and taking the supremum over unit vectors proves the bound. It implies $\|[A,B]\|\leq2\|A\|\|B\|$ by the <triangle inequality>.

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