Equivalent norms on space of ultra-distribution

Question

In the third volume of the book "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes it is stated (Remark 1.2., page 16) that on the space $ \mathcal D _{M_k}(\Omega , \mathcal K , L)$ the norms $$ \|\varphi\|_{\infty,\infty} := \sup_{x \in \mathcal K, \alpha \in \mathbb N ^d} \frac{|\partial^\alpha \varphi(x)|}{L^kM_k},\quad \|\varphi\|_{\infty,2} := \sup_{\alpha \in \mathbb N ^d} \frac{\|\partial^\alpha \varphi\|_{L^2(\mathcal K)}}{L^kM_k},\quad \|\varphi\|_{2,2} := \sqrt{ \sum_{k=0}^\infty \frac{\|\partial^\alpha \varphi\|_{L^2(\mathcal K)}}{L^{2k}M_k^2}} $$ are equivalent. This I fail to convince myself and I would like to understand it in the simplest case that matches their framework: $$ \Omega = \mathbb R,\quad \mathcal K = [0,1],\quad L = 1,\quad M_k = (k!)^2 $$ which means that the set $V := \mathcal D _{M_k}(\Omega , \mathcal K , L)$ is made of those $\varphi \in C^\infty[0,1]$ such that $$ \forall k \in \mathbb N,\quad \varphi^{(k)}(0) = \varphi^{(k)}(1) = 0 $$ and $$ \sup_{k \in \mathbb N} \frac{\| \varphi^{(k)}\|_{L^\infty(0,1)}}{(k!)^2} < \infty. $$ It is trivial that $$ \| \cdot \|_{\infty , 2} \lesssim \| \cdot \|_{\infty , \infty},\quad \| \cdot \|_{\infty , 2} \lesssim \| \cdot \|_{2,2} $$ but for instance I cannot see why $$ \| \cdot \|_{\infty , \infty} \lesssim \| \cdot \|_{\infty , 2}. $$ Of course I should use the Poincaré inequality $$ \exists C_P > 0,\quad \forall u \in H^1_0(0,1),\quad \|u\|_{L^\infty(0,1)} \leq C_P \|u'\|_{L^2(0,1)} $$ but it only gives me that for all $\varphi \in V$, $x \in [0,1]$ and $k \in \mathbb N$, $$ \begin{align*} \frac{|\varphi^{(k)}(x)| }{(k!)^2} &\leq C_P\frac{\|\varphi^{(k+1)}\|_{L^2(0,1)} }{(k!)^2} \\ &= C_P \frac{((k+1)!)^2}{(k!)^2} \frac{\|\varphi^{(k+1)}\|_{L^2(0,1)} }{((k+1)!)^2} \\ &= C_P (k+1)^2 \frac{\|\varphi^{(k+1)}\|_{L^2(0,1)} }{((k+1)!)^2} \end{align*} $$ Any reference, hint or counter-exampe would be appreciated.