PlanetMath: quotient norm

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quotient norm

(Definition)

Let $V$ be a normed vector space with norm $\| \cdot \|$ . Let $M$ be a closed subspace of $V$ and $V/M$ the quotient vector space.

The norm $\| \cdot \|$ induces a norm $\| \cdot \|_{V/M}$ in $V/M$ , called the quotient norm, given by

$\displaystyle \Vert v+M \Vert _{V/M}:= \inf_{u \in v+M} \Vert u\Vert = \inf_{m \in M} \Vert v+m\Vert$

Theorem - $\| \cdot \|_{V/M}$ is a norm in $V/M$ iff $M$ is closed in $V$ .

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Cross-references: iff, theorem, induces, quotient vector space, subspace, closed, norm, normed vector space
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This is version 2 of quotient norm, born on 2007-07-07, modified 2007-07-07.
Object id is 9749, canonical name is QuotientNorm.
Accessed 1558 times total.

Classification:

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

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