Definition:Distance Functional

Definition

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 $x$ if and only if:

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

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

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

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $20.2$: The Distance Functional