Estoy investigando los subconjuntos cerrados y acotados de $\ell_1$ con la norma de $|\cdot|_1$. Para el concreteness, $$\ell_1 = \left{x = (x_1,x_2,\dots)\ |\ |x|1 = \sum{i=1}^{\infty} |x_i|
Para todos los $n \in \mathbb{N}$, definir la secuencia $$\tilde{e}_n = (0,\dots,0,1 + 1/n,0,\dots).$$ That is, the $i$-th entry of $\tilde{e}_n$ is $0$ if $i \neq n$ and $1+1/n$ if $i = n $. Then set $A = {\tilde{e}_1,\tilde{e}_2,\dots} \subset \ell_1.$ Clearly $A$ is bounded, but is it closed? I believe it is vacuously closed since $A$ does not have any limit points. My end goal here is to exhibit a closed and bounded subset $A$ of $\ell_1$ such that the continuous function $f: \ell_1 \rightarrow \mathbb{R}$ defined by $f(x) = \sum_i x _I $ does not achieve its infimum on $A$.
