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I am totally stuck at the following problem...

I need to find a sequence of linear functionals on $L^{\infty}[0,1]$ of norm 1 and converging to zero pointwise. That is, $\{f_n\}$ in the dual of $L^{\infty}[0,1]$ such that each $f_n$ is of norm 1 and $f_n(x)$ goes to zero for each element $x$ of $L^{\infty}[0,1]$.

I tried everything I can think of. I tried to find $f_n$ in terms of integral w.r.t $L^1$ functions, tried to define them on the continuous functions and extend to $L^{\infty}[0,1]$. However nothing works....And it is so extremely frustrating.... Is there some magic way to solve this problem? Could anyone please help me?

Keith
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    Since $L^1 \subset (L^{\infty})^*$, you can just find a sequence of $L^1$ functions with norm $1$ that converge pointwise to zero... – Aphelli Sep 22 '19 at 18:53
  • No, the pointwise convergence of $L^1$ function does not guarantee the pointwise convergence of the correspoonding linear functional on $L^{\infty}[0,1]$....That is why I am stuck – Keith Sep 22 '19 at 19:01
  • please...no help?............. – Keith Sep 22 '19 at 19:06
  • Right... I didn’t think of that. Actually, thinking about it, and comparing with the $\ell$ situation, it is even possible that weak-* convergence in $(L^{\infty})^*$ for $L^1$ functions implies convergence in $L^1$... – Aphelli Sep 22 '19 at 19:26
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    Oh it is the Fourier coefficients.....I now see that – Keith Sep 22 '19 at 19:38
  • Well, that is a quite nice example. – Aphelli Sep 22 '19 at 20:15

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