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In general, do the notation $ f \in C^0 ([0,T],X) $ imply that $\| f(t) \|_X < \infty$ for any fixed $t \in [0,T]$? Or just means $\| f (t) \|_X$ is continuous on $[0,T]$? Here $X$ is like $L^p$ space or Sobolev space.

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    Possibly it means just continuous, unlike, for example, $C^1$ which means continuously differentiable?2012-10-21

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If $X$ is any sensible space, i.e. a Banach space, then a continuous $f$ sends the compact interval $[0,T]$ to a compact set, which is bounded, hence the $X$-norm of $f(t)$ is always finite.

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    Thank you. I think that this notation means \| f(t) \|_X < \infty for all $t \in [0,\infty)$ and $\| f(t) \|_X$, the function of t, is continuous on $[0,\infty)$. Is this suitable?2012-10-21