Exponentially weighted supremum norm ($\Vert u{\Vert }_{\alpha }={\sup }_t{e}^{-\alpha t}\Vert u(t)\Vert$)

On continuous functions $u:[0,T]\to X$, the exponentially weighted supremum norm is $\Vert u{\Vert }_{\alpha }={\sup }_{0\le t\le T}{e}^{-\alpha t}\Vert u(t)\Vert$. It is equivalent to the ordinary supremum norm and often turns a Volterra integral map into a contraction mapping.