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find a continuous functional $f \in (\ell^{1})^{*}$ so that $M \subseteq \ker f$

Note $M:=\{x \in \ell^{c}: \sum\limits_{n \geq 1}x_{n}=0\}$ and

$\ell^{c}=\{x \in \ell^{\infty}:x_{n}=0 \operatorname{for all but finitely many} n \in \mathbb N\}$

My only idea:

setting $f(x)=\sum\limits_{n \geq 1}x_{n}$ and we know for $x \in M$ that $\sum\limits_{n \geq 1}x_{n}=0$ thus $M\subseteq \ker f$

Is this correct?

Why does this suffice to show that $M$ is not dense in $\ell^{1}$

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What you have done is correct. For the last part note the kernel of any continuous linear functional is closed. Hence, if $M$ is dense then it must be equal to $\ell^{1}$. This is certainly false, right?

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