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Let $X_0=(X,\|\cdot\|_0)$ be a Banach space and $X_1=(X,\|\cdot\|_1)$ a normed vector space (not complete) such that there is a constant $c>0$ such that \begin{align*} \|\cdot\|_1 \leq c\|\cdot\|_0 \end{align*} Now given an operator $T$ satisfying for some $\alpha >0$ and $p>1$

\begin{align*} \int_0^\alpha \|(Tf)(t)\|_1^pdt \leq M\|f\|_{L^p([0,\alpha],X_0)} \end{align*} for some $M>0$ and all $f\in L^p([0,\alpha],X_0)$.

It is to be noted that, in my knowledge, we cannot define the space $L^p([0,\alpha],X_1)$ since $X_1$ is not complete. Now my question is : can I see that $T\in \mathcal{L}(L^p([0,\alpha],X_0))$?

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  • $\begingroup$ The estimate suggests only $T\in \mathcal(L^p(X_0), L^p(X_1))$ - if the latter space would make sense. $\endgroup$ – daw Mar 27 at 20:41
  • $\begingroup$ Yes, you're right. I was thinking if I can obtain a result juste like the one obtained here link (first lemma). $\endgroup$ – hitchcock Mar 27 at 22:04

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