Answer to Question #122935 in Real Analysis for Ruksan

Question

On R^n show that || . ||∞ ≤ || . ||2

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Answer 1

Let,$x\in \mathbb{R}^n$ such that $x=(x_1,\dots,x_n)$ ,thus

$$||x||_2=\sqrt{\sum_{k=1}^{n}|x_i|^2}$$

And,

$$||x||_{\infty}=\max_{1\leq i \leq n}\{|x_i|\}$$

Let, WLOG , $||x||_{\infty}=x_k$ for some $k\in\{1,\dots,n\}$ but also note that

$$||x||_{\infty}^2=x_k^2\leq x_1^2+\dots+x_k^2+\dots+x_n^2=||x||_2^2\\ \iff ||x||_{\infty}\leq ||x||_2$$

Hence, we are done.