# Bochner integral on non complete space

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 $$T\in \mathcal{L}(L^p([0,\alpha],X_0))$$?

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