# what is the infinity subscript mean in a real number?

I have come across an inequation in an optimization paper.

$$\left\lVert x\right\rVert _{2} \leq G$$ $$\left\lVert x\right\rVert _{\infty} \leq G_{\infty}$$

I know that the euclidean norm of a vector is bound by some real number $$G$$, but what does it mean in the second inequation? what is the $$G_{\infty}$$ mean?

The norm in the second relation is the uniform (maximum or supremum) norm, usually defined on bounded functions defined in some domain, say, $$D$$. If $$f$$ is continuous, then $$\|f\|_\infty=\lim_{p\rightarrow\infty}\|f\|_p$$ where $$\|f\|_p$$ is the $$p$$-norm. Since for each $$p$$, $$\|f\|_p\leq G_p$$ for some $$G_p$$, $$G_\infty$$ is the notation for the limit of $$\{G_p\}$$ as $$p\rightarrow\infty$$.
If the Euclidean norm $$\|x\|_2$$ is bounded by $$G$$, then every norm of $$x$$ is bounded by some $$G’$$. So I guess it means that $$G_\infty$$ is the corresponding bound of $$\|x\|_\infty$$.
• $$||x||_\infty=\lim_p||x||_p=\max_i |x_i|$$ Aug 17, 2019 at 8:15