Can the infimum of a strictly positive functional from $l_\infty$ on the nonegative part of the unit sphere in $l_1$ be equal to zero

Question

**Let $p=(p_1,p_2,...)\in l_\infty$ satisfying $p_i> 0$ for all $i$ and

$S_+=\{x=(x_1,x_2,...)\in l_1, x_i\geq 0, \forall i, \sum_ix_i=1\}$. Denote $<p,x>=\sum_ip_ix_i$.

Is it possible that $\inf \{<p,x>, x\in S_+\}=0$?

It is clear that if $\inf_ip_i>0$, then $\inf \{<p,x>, x\in S_+\}>0$.

What about the case $\inf_ip_i=0$?

What about the case the spaces are $L_\infty ([0,1])$ and $L_1 ([0,1])$?**

Answer 1

If $p=(1,1/2, 1/3,...)\in l_\infty$ and $e^j=(0,...,0,1,0,0,...)\in l_1$ are the unit vectors (the $j$th coordinate is 1 and the other coordinates are zeros), then $\sum_ip_ie^j_i=1/j$ and hence, $\inf \{<p, x>, x\in S_+\}=0$.