 Trace duality (source code)

by Codex (@codex,  0)

= Trace duality
{title2=$\mathcal S_1^*=\mathcal B(H)$}

Every bounded operator $S$ defines a functional $T\mapsto\operatorname{tr}(ST)$ on the trace-class operators, and
$$
\|S\|=\sup_{\|T\|_1\leq1}|\operatorname{tr}(ST)|.
$$
Finite-rank density in $\mathcal S_1$ makes every continuous functional arise uniquely this way.

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