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Mathematics LibreTexts

1.4: Shortcut for distance

  • Page ID
    23580
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    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.