quotient norm

Let $V$ be a normed vector space with norm $\parallel \cdot \parallel$. Let $M$ be a closed subspace of $V$ and $V/M$ thequotient vector space.

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

$${\parallel v+M\parallel }_{V/M}:=\underset{u\in v+M}{inf}\parallel u\parallel =\underset{m\in M}{inf}\parallel v+m\parallel$$

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