4.5 KiB
4.5 KiB
aliases, up, tags, excalidraw-plugin, excalidraw-open-md
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[!definition] Définition Soit
(X, d)
un espace métrique SoientA, B \subset X
deux parties deX
On appelle distance entreA
etB
: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
A
etB
$= "![[" + dv.current().file.name + ".svg|800]]"
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 B
et d'écrire :d(A, B) \leq d(x, x) = 0
D'où suit, par positivité des distances
Exemples
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Excalidraw Data
Text Elements
A ^rBDgNyV5
Drawing
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