Consider $D$ a set and $X$ a linear space of functions $D \rightarrow \mathbb{R}$ s.t. for any $f \in X$ and $g: D \rightarrow \mathbb{R}$, if $\lvert g \rvert \leq \lvert f \rvert$ then $g \in X$. Such spaces naturally arise as classes of asymptotic behaviors, for example if $D = \mathbb{N}$ and $h: \mathbb{N} \rightarrow \mathbb{R}$, we can take $X := \{f: \mathbb{N} \rightarrow \mathbb{R} \mid \lvert f \rvert = O(h)\}$. If $D$ has extra structure we can restrict consideration to functions satisfying certain conditions e.g. if $D$ is a topological space we can consider only continuous functions.

Is there a name for $X$ like that?

  • 3
    $\begingroup$ I don't know an "official" name, but I might call it "downward closed". $\endgroup$ – Nate Eldredge Feb 28 '16 at 18:32

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