Existence of Distance Functional

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 $Y$ be a proper closed linear subspace 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 $x$.