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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.

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.

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