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?
Best Answer
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 $p\rightarrow\infty$.