| dbp:proof
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- A proof can be sketched as follows:
Let be the weak*-compact set of positive linear functionals on with norm ≤ 1, and be the continuous functions on .
can be viewed as a closed linear subspace of . By Hahn–Banach, extends to a in with .
Using results from measure theory quoted above, one has:
where, by the self-adjointness of , can be taken to be a signed measure. Write:
a difference of positive measures.
The restrictions of the functionals and to has the required properties of and .
This proves the theorem. (en)
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