PlanetMath: scaling of the open ball in a normed vector space

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scaling of the open ball in a normed vector space

(Theorem)

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

$\displaystyle B_r(v) = \{ w\in V: \Vert w-v\Vert < r \}.$

Then for any $\lambda\in F$ , we have

$\displaystyle \lambda B_r(v) = B_{\vert\lambda\vert r}(\lambda v).$

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

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Cross-references: clear, norm, complex, real, field, vector space
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This is version 3 of scaling of the open ball in a normed vector space, born on 2005-10-29, modified 2006-10-14.
Object id is 7458, canonical name is ScalingOfTheOpenBallInANormedVectorSpace.
Accessed 949 times total.

Classification:

AMS MSC:

46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)

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