Ivan Shishkin, Rye (1878)

Problems/TopologyExerciseUnreviewed

Equivalence des 33 normes usuelles sur Rn\mathbb{R}^n

by SalixBabylonica·
40
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  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
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Soit n∈N∗n\in\mathbb N^* et X=(x1⋮xn)∈Rn.X= \begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix} \in\mathbb R^n.
On définit :
∥X∥1=∑k=1n∣xk∣,\|X\|_1 = \sum_{k=1}^{n}|x_k|,∥X∥∞=max⁡1≤k≤n∣xk∣,\|X\|_\infty = \max_{1\leq k\leq n}|x_k|,∥X∥2=∑k=1n∣xk∣2.\|X\|_2 = \sqrt{\sum_{k=1}^{n}|x_k|^2}.Ces trois applications sont des normes sur Rn\mathbb R^n.
Montrez que ces normes sont équivalentes sur Rn\mathbb R^n.

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