It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$ ||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the norms are taken over space (after fixing time). Is there a name for this inequality?
EDIT: the proof is \begin{align*} |u(x,t)| &= \left| \int_0^x u_y(y,t) dy \right| \\ &\leq \int_0^x \left| u_y(y,t) \right| dy \\ &\leq \int_0^1 \left| u_x(x,t) \right| dx \\ &\leq \left( \int_0^1 1^2 dy \right)^{1/2} \left( \int_0^1 |u_x(x,t)|^2 dx \right)^{1/2} \\ &= \left( \int_0^1 |u_x(x,t)|^2 dx \right)^{1/2} \end{align*}
Any references would be helpful.