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I want to find a discontinuous linear map $\phi: c_0 \to \mathbb{C}$. where $c_0$ has sup norm obviously, $\|.\|_\infty$

I can't think of any example. please suggest me one. I ll try checking it myself.

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Abstract solution (apparently it is not possible a concrete example): take a Hamel basis of $c_0$, $(b_i)_{i\in I}$ with $\Bbb N\subset I$ and $\lim_{n\to\infty}\|b_n\|=0$ (why this is possible?).

For each $$x=\sum_{i\in I}\lambda_i b_i\in c_0$$ take $$\phi(x)=\sum_{i\in I}\lambda_i.$$ The sequence $(b_n)_{n\in\Bbb N}\to 0$ but $$\lim_{n\to\infty}b_n=\lim_{n\to\infty}1=1.$$

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