Category:Existence of Distance Functional
Jump to navigation
Jump to search
This category contains pages concerning Existence of Distance Functional:
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 $x$.
Pages in category "Existence of Distance Functional"
The following 3 pages are in this category, out of 3 total.