I have a question: if $f$ is uniformly bounded in $L^2(0,T,X)$ , then $f$ is uniformly bounded a.e. in $X \times (0,T).$ If yes, how to prove it? Thank you.
2026-04-28 17:46:13.1777398373
Function bounded a. e.
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It's not true even for classical $L^2$ space (then the counter-example can be generalized). Take $f_n:=\sqrt n \chi_{(0,1/n)}$: this sequence is bounded in $L^2$, but not in $L^\infty$.