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