\begin{definition}[Definition:Distance Functional]
Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
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 $Y$.
We say that $f \in X^\ast$ is a '''distance functional for $x$''' {{iff}}: 
:$(1): \quad$ $\norm f_{X^\ast} = 1$
:$(2): \quad$ $\map f y = 0$ for each $y \in Y$
:$(3): \quad$ $\map f x = d$.
\end{definition}