Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 1 / b / v / Solution

True. For every finite partial sum,The nondecreasing partial sums on the left are bounded, so the square series converges. Hence ${\ell }^1\subseteq {\ell }^2$ andApplying this norm inequality to $x-y$ proves $\Vert f(x)-f(y){\Vert }_2\le \Vert x-y{\Vert }_1$. Thus the inclusion is a Lipschitz continuous map, in particular continuous, and it is well-defined on every element of l1 sequence space.