Definition:Quotient Norm

Definition

Let $X$ be a normed vector space.

Let $N$ be a closed linear subspace of $X$.

Let $X/N$ be the quotient vector space of $X$ modulo $N$.

Let $\pi : X \to X/N$ be the quotient mapping associated with $X/N$.

We define the quotient norm $\norm \cdot_{X/N}$ by:

$\ds \norm {\map \pi x}_{X/N} = \inf_{z \in N} \norm {x - z}$

for each $x \in X$.

Also see

• Definition:Normed Quotient Vector Space
• Quotient Norm is Norm shows that $\norm \cdot_{X/N}$ is well-defined and a norm
• Topology Induced by Quotient Norm is Quotient Topology

• Results about quotient norms can be found here.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.41$: Theorem