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Suppose that $m \in \mathbb{N}$ and $\mathbb{R}^m=\{(x_1, ..., x_m)|x_i\in \mathbb{R}\}$ is the real vector space of $m$-tuples of real numbers. Let $\|\cdot\|:\mathbb{R}^m \to [0, \infty)$ be a norm on $\mathbb{R}^m$.

A function $f:\mathbb{R}^m \to \mathbb{R}$ is called absolutely homogeneous if $f(tx)=|t|f(x)$ for every $t \in \mathbb{R}$ and $x \in \mathbb{R}^m$. Let $f:(\mathbb{R}^m, \|\cdot\|) \to (\mathbb{R}, |\cdot|)$ be a continuous function with $f(x)>0$ for every non-zero $x \in \mathbb{R}^m$. Show that if $f$ is absolutely homogeneous then there exists a positive constant $\alpha >0$ such that $\alpha \|x \| \leq f(x)$ for every $x \in \mathbb{R}^m$.

metamorphy
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Alex
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  • Use a descriptive title 2) What have you tried?/Where are you stuck?/What is your question? Consider editting your question to reflect these 2 points so as to avoid getting downvoted
    • – Stefan Octavian Jan 11 '21 at 16:58
  • I've just started learning this area, so don't really know how to solve this question, like what theorem or definitions should I use here. – Alex Jan 11 '21 at 17:08
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    Hints : Why is this function "determined" by the set ${|x| = 1}$? Why is that set compact? So $f$ attains a minimum and maximum on that set. Can you conclude now? – Sarvesh Ravichandran Iyer Jan 11 '21 at 18:14
  • I've proved that the set ${ x \in R^m: |x|=1}$ is compact in $(R^m, | \cdot |)$. Moreover, I know that I need to use the extreme value theorem here, but I don't know how exactly to apply it. Could you please give me a more detailed explanation. Thank you! – Alex Jan 11 '21 at 19:35