Existence of Distance Functional/Proof 2

Theorem

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

Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\mathbb F$.

Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $X$.

Let $x \in X \setminus Y$.

Let:

$d = \map {\operatorname {dist} } {x, Y}$

where $\map {\operatorname {dist} } {x, Y}$ denotes the distance between $x$ and $Y$.

Then there exists $f \in X^\ast$ such that:

$(1): \quad$ $\norm f_{X^\ast} = 1$

$(2): \quad$ $\map f y = 0$ for each $y \in Y$

$(3): \quad$ $\map f x = d$.

That is:

there exists a distance functional for $x$.

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:

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

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

with:

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

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

$\blacksquare$