Bounded continuous functions
= Bounded continuous functions
{title2=$C_b(Y),\quad\|f\|_\infty=\sup_Y|f|$}
<Bounded continuous functions> on a topological space form a <Banach space> with the <supremum norm>. A uniform limit is continuous and bounded, proving completeness. This space differs from the bounded uniformly <continuous functions>: free-transport composition is jointly continuous in time and position, but need not be continuous in the global sup <norm> as time varies.