Can somebody please help me to find the answer for this problem...
Let $V$ be a norm linear space and let $x\in V\setminus\{0\}$. Also let $W$ be a linear subspace of $V$. Show that if there is $r>0$ such that $\{y \in V \mid \lVert y\rVert< r\}$ is a subset of $W$, then $\frac{rx}{2\lVert x\rVert}\in W$.