Riesz's Lemma

Riesz's Lemma

Let $Y$ be a proper closed subspace of a normed vector space $X$ cover $\mathbb{R}$. Then for each $\alpha <1$, there exists $x\in X$ with $\Vert x\Vert =1$ such that $d(x,Y)={\inf }_{y\in Y}\Vert x-y\Vert >\alpha$

Proof Take $z\in X$, $z\notin Y$. Since $Y$ closed, we have $d(z,Y)>0$. By rescaling we can assume $d(z,Y)=1$. Then for $0<\alpha <1$, there exists $y\in Y$ such that $1\le \Vert z-y\Vert <\frac{1}{\alpha }$. Now take $x=\frac{z-y}{\Vert z-y\Vert }$ so that $\Vert x\Vert =1$ and $d(x,Y)=\frac{d(z-y,Y)}{\Vert z-y\Vert }=\frac{1}{\Vert z-y\Vert }>\alpha$.