1.4: Shortcut for distance
Most of the time, we study only one metric on space. Therefore, we will not need to name the metric each time.
Given a metric space \(\mathcal{X}\), the distance between points \(A\) and \(B\) will be further denoted by
\(AB\) or \(d_{\mathcal{X}}(A,B)\);
the latter is used only if we need to emphasize that \(A\) and \(B\) are points of the metric space \(\mathcal{X}\).
For example, the triangle inequality can be written as
\(AC \le AB + BC\).
For the multiplication, we will always use "\(\cdot\)", so \(AB\) could not be confused with \(A \cdot B\).
Exercise \(\PageIndex{1}\)
Show that the inequality
\(AB + PQ \le AP + AQ + BP + PQ\)
holds for any four points \(A, B, P, Q\) in a metric space.
- Hint
-
Sum up four triangle inequalities.