I'm studying by myself the the chapter of second order parabolic linear equations by Evan's book, which focus in solve
$$(11) \ \begin{cases} \begin{eqnarray*} u_t + Lu &=& f \ \text{in} \ U_T\\ u &=& 0 \ \text{on} \ \partial U \times [0,T]\\ u &=& g \ \text{on} \ U \times \{ t = 0 \} \end{eqnarray*} \end{cases}$$
I'm trying understand how the theorem $7$ on page $388$ is a direct consequence of the theorem $6$ on page $386$. Specifically, I'm trying understand how the following theorem:
$\textbf{Theorem 7.}$ (Infinite differentiability). Assume
$$g \in \mathcal{C}^{\infty}(\overline{U}), f \in \mathcal{C}^{\infty}(\overline{U}_T),$$
and the $m^{th}$-order compatibility conditions hold for $m = 0, 1, \cdots$.
Then the parabolic initial/boundary value-problem $(11)$ has an unique solution
$$u \in \mathcal{C}^{\infty}(\overline{U}_T).$$
is a direct consequence of this theorem:
$\textbf{Theorem 6.}$ (Higher Regularity). Assume
$$\begin{cases} g \in H^{2m+1}(U),\\ \frac{d^kf}{dt^k} \in L^2(0,T;H^{2m-2k}(U)) \ (k = 0, \cdots, m). \end{cases}$$
Suppose also that the following $m^{th}$-order compatibility conditions hold
$$\begin{cases} g_0 := g \in H^1_0(U), g_1 := f(0) - L g_0 \in H^1_0(U),\\ \cdots, g_m := \frac{d^{m-1}f}{dt^{m-1}}(\cdot, 0) - Lg_{m-1} \in H^1_0(U). \end{cases}$$
Then
$$\frac{d^k\textbf{u}}{dt^k} \in L^2 (0,T;H^{2m+2-2k}(U)) \ (k = 0, \cdots, m+1),$$
and we have the estimate
$$(55) \ \sum_{k=0}^{m+1} \left| \left| \frac{d^k\textbf{u}}{dt^k} \right| \right|_{H^{2m+2-2k}(U)} \leq C \left( \sum_{k=0}^{m} \left| \left| \frac{d^k\textbf{f}}{dt^k} \right| \right|_{L^2(0,T;H^{2m-2k}(U))} + ||g||_{H^{2m+1}(U)} \right),$$
the constant $C$ depending only on $m, U, T$ and the coefficients of $L$.
Evans said simply that the theorem $7$ follows from the theorem $6$ applying this theorem for $m = 0, 1, 2, \cdots$, but I didn't understand how the bounds for the higher derivatives of $\textbf{u}$ with respect for the norm of $L^2(0,T;H^{2m+2-2k}(U))$ imply that these same higher derivatives are bounded with respect to the uniform norm over $\overline{U}_T$, which imply that $u \in \mathcal{C}^{\infty}(\overline{U}_T)$. I would like to understand it.
Thanks in advance!
P.S.: just to make the question self-contained, I will introduce some definitions used on theorems $6$ and $7$.
$\textbf{Definition.}$ Let be $X$ a real Banach space with norm $|| \ ||$. The space $L^p(0,T;X)$ consists of all strongly measurable functions $\textbf{u}: [0,T] \longrightarrow X$ with
$$(i) \ ||\textbf{u}||_{L^p(0,T;X)} := \left( \int_0^T ||\textbf{u}(t)||^p \right)^{\frac{1}{p}} < \infty$$
for $1 \leq p < \infty$.
$\textbf{Definition.}$ Let be $U$ an open set of $\mathbb{R}^n$, then $H^k(U) := W^{k,2}(U)$.