Question:
Let $X=C[0,1]$, show that there is no such norm $\lVert\cdot\rVert_*$ on $X$ that for any series $\{f_n\}_{n=1}^{\infty}\subset X$,
$\lim_{n\to\infty}\lVert f_n\rVert_*\to 0\Longleftrightarrow \lim_{n\to\infty}f_n(t)=0,\quad\forall t\in[0,1]$
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I've tried to define a new norm (supposing such $\lVert\cdot\rVert_*$ exists): $\lVert f\rVert_+=\lVert f\rVert_*+\max_{t\in[0,1]}|f(t)|=\lVert f\rVert_*+\lVert f\rVert_C$
it is easy to show that $\lVert\cdot\rVert_+$ is a complete norm on $X$ (so is $\lVert\cdot\rVert_C$), and this implies $\lVert\cdot\rVert_+$ and $\lVert\cdot\rVert_C$ are equivalent norms, so there is a constant $M$ s.t.
$\lVert f\rVert_*\leq M\lVert f\rVert_C,\quad f\in X$
and I got stuck at the above inequality (or maybe it is useless).