# quotient norm

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

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

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

Title quotient norm QuotientNorm 2013-03-22 17:22:58 2013-03-22 17:22:58 asteroid (17536) asteroid (17536) 5 asteroid (17536) Definition msc 46B99