For $r,c>0$ put $$X_{c,r}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r, \, \forall i\in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_{c, r}} \|x\|_2=0$. Is it possible to compute $$s_c:=\sup_{x \in X_{c,r}} \|x\|_2 \quad ?$$ Does at least $\lim_{c \to 0} s_c=0$ hold?
2026-04-05 15:44:14.1775403854
Maximum of $\ell^2$-Norm
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For you very last question, notice that if $x\in X_{c, t}$, $$\|x\|^2_2=\sum_nx^2_n<c\sum_n|x_n|=c\|x\|_1=cr$$ Thus $s_{c, r}\leq \sqrt{cr}\xrightarrow{c\rightarrow0}0$.