Bounded continuous functions ($C_b(Y),\Vert f{\Vert }_{\infty }={\sup }_Y|f|$)

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.