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I have a set $\mathcal{F}$ of real-valued functions, $$f_i(\cdot):\mathbb{R}\to\mathbb{R} \, ,$$ and a (linear) functional $T$ defined on $\mathcal{F}$, $$T:f_i \mapsto T[f_i] \in \mathbb{R} \, ,$$ such that $$f_1(x)\geq f_2(x) \ \ \forall x \in \mathbb{R} \Rightarrow T[f_1] \geq T[f_2]\, .$$

Is there a standard term for this property? Can I call $T$ 'monotonic,' or 'pointwise monotonic'? (In particular, 'pointwise increasing'?)

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When $\mathcal F $ is a subspace and $T$ linear it called a positive linear form. as example $T(f)=\int_a^bf(t)dt$ with $a<b$ and $\mathcal F=\mathcal C([a,b],\mathbb R)$.

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  • $\begingroup$ Thanks!. Oops, I realize my $T$ is affine, not linear. Anyway this is close enough. I guess I can call the offset-subtracted part a positive linear functional. $\endgroup$ – GrayOnGray Aug 21 '14 at 23:20

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