5.1 KiB
5.1 KiB
aliases, up, tags, excalidraw-plugin, excalidraw-open-md
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[!definition] Définition Soit
(X, d)un espace métrique SoientA, B \subset Xdeux parties deXOn appelle distance entreAetB:d(A, B) := \inf\limits \{ d(a, b) \mid a \in A \wedge b \in B \}^definition
[!idea] intuition Cette distance représente normalement la plus courte distance entre
AetB!distance entre deux parties d'un espace métrique.svg
Propriétés
[!proposition]+ distance entre des parties non disjointes Si
A \cap B \neq \emptyset, alorsd(A, B) = 0[!démonstration]- Démonstration Il suffit de prendre
x \in A \cap Bet d'écrire :d(A, B) \leq d(x, x) = 0D'où suit, par positivité des distances
Exemples
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Excalidraw Data
Text Elements
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B ^kynULWDT
Drawing
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