How to prove that an positive linear operator $T:C[0,1]\to R $ in the sense that $T(f)\geq 0$ when $f\geq 0$ is bounded?
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Suppose $\|f\|_\infty \leq 1$. Then $-1\le f\le 1$ so $-T(1) \le T(-1)\le T(f) \le T(1)$ so $\|T\| \le T(1)$. In fact equality is achieved, since $\|1\|_\infty = 1$.