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\|<r\}.

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

$\displaystyle\lambda B_{r}(v)$	$\displaystyle=$	$\displaystyle\{z\in V:\\|w-v\\|<r\ \mbox{and}\ z=\lambda w\}$
	$\displaystyle=$	$\displaystyle\{z\in V:\\|\frac{z}{\lambda}-v\\|<r\}$
	$\displaystyle=$	$\displaystyle\{z\in V:\\|z-\lambda v\\|<\|\lambda\|r\}$
	$\displaystyle=$	$\displaystyle B_{\|\lambda\|r}(\lambda v).$

Title	scaling of the open ball in a normed vector space
Canonical name	ScalingOfTheOpenBallInANormedVectorSpace
Date of creation	2013-03-22 15:33:25
Last modified on	2013-03-22 15:33:25
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	7
Author	matte (1858)
Entry type	Theorem
Classification	msc 46B99