Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let $T\in (0,\infty)$. Assume that \begin{equation} u\in L^2((0,T),V)\cap H^1((0,T),V'),\qquad f\in L^2((0,T),V')\,. \end{equation} I wonder if I can infer \begin{equation} u\in C((0,T),V)\cap C^1((0,T),V'),\qquad f\in C((0,T), X)\,. \end{equation}
Indeed, I am trying to adapt Theorem 11.3 of M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, Springer New York, NY, (2004).
Ultimately, using Lemma 11.4 of the same book, I would like to have \begin{equation} u\in C([0,T],H)\cap C((0,T),V)\cap C^1((0,T),V')\,. \end{equation}
But I am interested in the idea behind such a setting, that is why I started thinking of Lion's theorem and I also found Theorem 11.3 of M. Renardy, R.C. Rogers book. I wonder if I can get my desired result using the mentioned theorem? or they are not related due to having a different regularity in time. More precise question can be found here.
