Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1

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:

$\norm \pi_{\map B {X, X/N} } = 1$

where $\norm {\, \cdot \,}_{\map B {X, X/N} }$ denotes the norm on the space of bounded linear transformations.

Proof

Let $B_X$ be the unit open ball in $\struct {X, \norm {\, \cdot \,} }$.

Let $B_{X/N}$ be the unit open ball in $\struct {X/N, \norm {\, \cdot \,} }$.

From Quotient Mapping is Bounded in Normed Quotient Vector Space, $\pi$ is bounded.

We have:

\(\ds \norm \pi_{\map B {X, X/N} }\)

\(=\)

\(\ds \sup_{x \in B_X} \norm {\map \pi x}_{X/N}\)

\(\ds \)

\(=\)

\(\ds \sup_{q \in \map \pi {B_X} } \norm q_{X/N}\)

\(\ds \)

\(=\)

\(\ds \sup_{q \in B_{X/N} } \norm q_{X/N}\)

Quotient Mapping Maps Unit Open Ball in Normed Vector Space to Unit Open Ball in Normed Quotient Vector Space

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$