Existence of Distance Functional/Proof 2

Proof
Consider the normed quotient vector space $X/Y$ with quotient mapping $\pi$.

From Kernel of Quotient Mapping, we have $\map \pi x \ne 0$.

So, from Existence of Support Functional, there exists $f \in \paren {X/Y}^\ast$ such that:


 * $\norm f_{\paren {X/Y}^\ast} = 1$

and:


 * $\map f {\map \pi x} = \norm {\map \pi x}_{X/Y}$

From the definition of the quotient norm, we have:


 * $\norm {\map \pi x}_{X/Y} = \map {\operatorname {dist} } {x, Y}$

From Normed Dual Space of Normed Quotient Vector Space is Isometrically Isomorphic to Annihilator, $g = f \circ \pi \in X^\ast$ and:


 * $\norm g_{X^\ast} = \norm f_{\paren {X/Y}^\ast} = 1$

with:


 * $\map g x = \map {\operatorname {dist} } {x, Y}$

So $g$ is a linear functional satisfying our requirements.