Past exam of the analysis course of the University of Oxford / 2011 / 1 / Solution

A normed vector space over $\mathbb{R}$ is a real vector space $X$ with a function $\Vert \cdot \Vert :X\to [0,\infty )$ satisfyingIts distance is $d(x,y)=\Vert x-y\Vert$. It is a Banach space if this metric is complete: every Cauchy sequence converges to an element of $X$.