# scaling of the open ball in a normed vector space

Let $V$ be a vector space over a field $F$ (real or complex), and let $\|\cdot\|$ be a norm on $V$. Further, for $r>0$, $v\in V$, let

 $B_{r}(v)=\{w\in V:\|w-v\|

Then for any non-zero $\lambda\in F$, we have

 $\lambda B_{r}(v)=B_{|\lambda|r}(\lambda v).$

The claim is clear for $\lambda=0$, so we can assume that $\lambda\neq 0$. Then

 $\displaystyle\lambda B_{r}(v)$ $\displaystyle=$ $\displaystyle\{z\in V:\|w-v\| $\displaystyle=$ $\displaystyle\{z\in V:\|\frac{z}{\lambda}-v\| $\displaystyle=$ $\displaystyle\{z\in V:\|z-\lambda v\|<|\lambda|r\}$ $\displaystyle=$ $\displaystyle B_{|\lambda|r}(\lambda v).$
Title scaling of the open ball in a normed vector space ScalingOfTheOpenBallInANormedVectorSpace 2013-03-22 15:33:25 2013-03-22 15:33:25 matte (1858) matte (1858) 7 matte (1858) Theorem msc 46B99