Uniform boundedness in $L^p$ norm

Question

Given a Banach space $X$ and a sequence $(f_n)_{n\in \mathbb{N}}\subset L^p([0,T];X)$ such that $$ \sup_{n\in \mathbb{N}} \Vert f_n \Vert_{L^p([0,T];X)} <+\infty $$ Can we say that $$ \sup_{n\in \mathbb{N}} \Vert f_n(t) \Vert_X <+\infty $$ for some (or every) $t\in [0,T]$? Thank you in advance.

Answer 1

Hint: Consider $1^{1/p}\chi_{[0,1]}, 2^{1/p}\chi_{[0,1/2]}, 2^{1/p}\chi_{[1/2,1]},3^{1/p}\chi_{[0,1/3]},3^{1/p}\chi_{[1/3,2/3]},3^{1/p}\chi_{[2/3,1]}, \dots$