TITLE: Terminology: 'pointwise monotone functional'?
QUESTION [0 upvotes]: 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'?)

REPLY [1 votes]: 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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