Let $l_{\infty}$ be the space of all bounded complex-valued sequences equipped with the supremum norm. Consider the natural standard basis $\{e_n\}_{n \in \mathbb{N}}$ of $l_{\infty}$. For any bounded linear functional $L \in {l_{\infty} }^* $, it is not difficult to see that $$ \sum_{n=1}^{\infty} f(n) L(e_n) = \sum_{n=1}^{\infty} L(f(n) e_n) $$ converges for any $f \in l_{\infty} $.

However, it seem that the above series does not necessarily converge to $$L(f)$$ since $\{ \sum_{k=0}^{n} f(k)e_k \}_n$ does not necessarily converges to $f$ in the supremum norm topology.

I wonder if there is any explicit example such that $$L(f) = \sum_{n=1}^{\infty} f(n) L(e_n) $$ does not hold.

It might be a standard example but I am afraid that I do not have yet concrete background in graduate-level real analysis or functional analysis so I cannot easily come up with such an example..

Thanks in advance.


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