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1.4: Shortcut for distance

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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.


    Sum up four triangle inequalities.

    This page titled 1.4: Shortcut for distance is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.