This is page 302 of PDE Evans, 2nd edition.
DEFINITIONS. $\text{(i)}$ The Sobolev space $$W^{1,p}(0,T;X)$$ consists of all functions $\textbf{u} \in L^p(0,T;X)$ such that $\textbf{u}'$ exists in the weak sense and belongs to $L^p(0,T;X)$. Furthermore, $$\|\textbf{u}\|_{W^{1,p}(0,T;X)} := \begin{cases} \left(\int_0^T \|\textbf{u}\|^p + \|\textbf{u}'(t)\|^p \, dt \right)^{1/p} & (1 \le p < \infty) \\ \operatorname{ess}\sup\limits_{0 \le t \le T} (\|\textbf{u}(t) \| + \| \textbf{u}'(t) \|) & (p=\infty).\end{cases}$$
THEOREM 2 (Calculus in an abstract space). Let $\textbf{u} \in W^{1,p}(0,T;X)$ for some $1 \le p \le \infty$. Then
$\text{(i)}$ $\textbf{u} \in C([0,T];X)$ (after possibly being redefined on a set of measure zero).
$\text{(ii)}$ $\textbf{u}(t)=\textbf{u}(s)+\int_s^t \textbf{u}' (\tau) \, d\tau$ for all $0 \le s \le t \le T$.
$\text{(iii)}$ Furthermore, we have the estimate $$\max_{0 \le t \le T} \|\textbf{u}(t) \|\le C\| \textbf{u} \|_{W^{1,p}(0,T;X)}, \tag{7}$$ the constant $C$ depending only on $T$.
Proof. 1. Extend $\textbf{u}$ to be $\textbf{0}$ on $(-\infty,0)$ and $T,\infty)$, and then set $\textbf{u}^\epsilon =\eta_\epsilon * \textbf{u}$, with $\eta_\epsilon$ denoting the usual mollifer on $\mathbb{R}^1$. We check as in the proof of Theorem 1 in §5.3.1 that $(\textbf{u}^\epsilon)' =\eta_\epsilon * \textbf{u}'$ on $(\epsilon, T-\epsilon)$.
$\quad$Then as $\epsilon \rightarrow 0$, \begin{cases}\textbf{u}^\epsilon =\eta_\epsilon * \textbf{u} & \text{in }L^p(0,T;X) \\ (\textbf{u}^\epsilon)' =\eta_\epsilon * \textbf{u}' & \text{in }L_{\text{loc}}^p(0,T;X). \tag{8} \end{cases} Fixing $0<s<t<T$, we compute $$\textbf{u}^\epsilon (t)=\textbf{u}^\epsilon (s) + \int_s^t \textbf{u}^{\epsilon '} (\tau) \, d\tau.$$ Thus $$\textbf{u}(t) = \textbf{u}(s)+\int_s^t \textbf{u}'(\tau) \, d\tau$$ for a.e. $0 < s < t < T$, according to $\text{(8)}$. As the mapping $t \mapsto \int_0^t \textbf{u}'(\tau) \, d\tau$ is continuous, assertions $\text{(i)}$, $\text{(ii)}$ follow.
$\quad$2. Estimate $\text{(7)}$ follows easily from $\text{(9)}$.
I understand 1. in the proof, but not 2. How exactly may I use $\displaystyle \textbf{u}(t)=\textbf{u}(s)+\int_s^t \textbf{u}'(\tau) \, d\tau$, over the interval $[0,T]$ (with $s,t$ in between said interval) to prove that $$\max_{0 \le t \le T} \|\textbf{u}(t) \|\le C\| \textbf{u} \|_{W^{1,p}(0,T;X)}$$ Relation $\text{(7)}$ resembles the many previous estimates shown in Chapter 5 of Evans' textbook. I have finally finished reading the entirety of Chapter 5, but are there also any recommended sections I should review as general background for this? (If this is not applicable here actually, then please disregard this last paragraph.)
First, you apply Hölder to get $$ \|u(t)\| \leq \|u(s)\| + \int_0^T\|u'(\tau)\|\,\mathrm{d}\tau \leq \|u(s)\| + T^{1-1/p}\|u'\|_{L^p(0,T;X)} . $$ Then, taking the $p$-th power of both sides, and integrating with respect to $s$, we have $$ T\|u(t)\|^p \leq c\int_0^T\|u(s)\|^p\,\mathrm{d}s + cT\cdot T^{p-1}\|u'\|_{L^p(0,T;X)}^p = c\|u\|_{L^p(0,T;X)}^p + cT^p\|u'\|_{L^p(0,T;X)}^p , $$ where $c=2^{p-1}$.