equibounded

Let $X$ and $Y$ be metric spaces. A family $F$ of functions from $X$ to $Y$ is said to be equibounded if there exists a bounded subset $B$ of $Y$ such that for all $f\in F$ and all $x\in X$ it holds $f(x)\in B$.

Notice that if $F\subset {𝒞}_b(X,Y)$ (continuous bounded functions) then $F$ is equibounded if and only if $F$ is bounded (with respect to the metric of uniform convergence).

Title	equibounded
Canonical name	Equibounded
Date of creation	2013-03-22 13:16:55
Last modified on	2013-03-22 13:16:55
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	5
Author	paolini (1187)
Entry type	Definition
Classification	msc 54E35
Synonym	equi bounded
Synonym	equi-bounded