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1 $begingroup$ Continuing from here Let $f_t(x):=f(x+t)$ Consider $f mapsto f_t$ which is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ for every $t in mathbb{R}^p$ How can I show that for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous? real-analysis analysis lp-spaces share | cite | improve this question asked Jan 5 at 13:56 user626880 user626880

2 2 $begingroup$ This question is related to this other post: I was wondering if proving that $ H trianglelefteq G$ and $Hcap G^{prime} ={e}$ (where $G^{prime}$ denotes the commutator subgroup of $G$) implies that the elements of $H$ commute with the elements of $G^{prime}$ is the same as proving that $H trianglelefteq G$ and $H cap G^{prime}={e}$ imply that $H subseteq C(G)$ (where $C(G)$ is the center of $G$)? If not, is there a way to modify/extend this proof to show that $H subseteq C(G)$? (It just needs to be a subset, not a subgroup). If still not, how can I prove that$H trianglelefteq G$ and $H cap G^{prime}={e}$ imply that $Hsubseteq C(G)$? Thank you. abstract-algebra group-theory normal-subgroups