what is the infinity subscript mean in a real number?

Question

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?

Answer 1

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 $\|x\|_\infty$.

Answer 2

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 $p\rightarrow\infty$.