Definition:Distance/Sets
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Definition
Metric Spaces
Let $V$ be a metric space with associated metric $d$.
Let $x \in V$, and let $S, T$ be subsets of $V$.
The distance between $x$ and $S$ is defined and annotated $\displaystyle d \left({x, S}\right) = \inf_{y \in S} \left({d \left({x, y}\right)}\right)$.
The distance between $S$ and $T$ is defined and annotated $\displaystyle d \left({S, T}\right) = \inf_{\substack{x \in S \\ y \in T}} \left({d \left({x, y}\right)}\right)$.
Real Numbers
Let $S, T$ be a subsets of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
The distance between $x$ and $S$ is defined and annotated $\displaystyle d \left({x, S}\right) = \inf_{y \in S} \left({d \left({x, y}\right)}\right)$, where $d \left({x, y}\right)$ is the distance between $x$ and $y$.
The distance between $S$ and $T$ is defined and annotated $\displaystyle d \left({S, T}\right) = \inf_{\substack{x \in S \\ y \in T}} \left({d \left({x, y}\right)}\right)$.
Alternative Notation
Some sources write $\operatorname{dist}$ instead of $d$.
Sources
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.13 \ (4)$
- John B. Conway: A Course in Functional Analysis (1990) $I.2.4$
