Quotient Mapping is Bounded in Normed Quotient Vector Space

Theorem

Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a normed vector space over $\Bbb F$.

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

Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the normed quotient vector space associated with quotient vector space $X/N$.

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

Then $\pi$ is a bounded linear transformation.

Proof

From Quotient Mapping is Linear Transformation:

$\pi$ is a linear transformation.

Let $x \in X$.

Note that from the definition of quotient norm, we have:

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

Note that since $0 \in N$, we have that:

$\norm x_X \in \set {\norm {x - z}_X : z \in N}$

so that:

$\ds \inf_{z \in N} \norm {x - z}_X \le \norm x$

from the definition of the infimum.

So we have:

$\norm {\map \pi x}_{X/N} \le \norm x$

for each $x \in X$, and so:

$\pi$ is a bounded linear transformation.

$\blacksquare$