How can I find a sequence that converges as for the norm but doesn't converge almost everywhere, in some space $L^p$ ??
Could you give me some hints ??
2026-03-25 07:49:03.1774424943
Sequence that converges as for the norm but not almost everywhere
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Hint:
Divide $[0,1]$ into $n$ subintervals with equal length, call each one $I_k^n$ where $1\leq k\leq n$, consider $f_{n+k}=\chi_{I_k^n}$, which is a sequence of function converges to $f=0$ in $L^1$ (since the length eventually converges to $0$) but nowhere converges (since $\forall x\in[0,1], x\in I_k^n$ for some $k$ for each $n$.)