Is there any result like the [Bramble-Hilbert lemma](https://en.wikipedia.org/wiki/Bramble%E2%80%93Hilbert_lemma) for Bochner spaces?


More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)$ for $Y$ a normed space. If $L$ vanishes on $k-1$ polynomials, do we necessarily have $||L v ||\_Y\leq C\_k ||\partial\_t^{k} v||\_{L^2(I,H)}$?



Approximation in Bochner spaces
Writing $$v(t)=p(t)+\frac{1}{(k-1)!}\int\_0^t (t-s)^{k-1}v^{(k)}(s)\, ds$$ with $p$ the Taylor polynomial at $0$ of degree $k-1$, then $\|v-p\|\_{H^k} \leq C\|v^{(k)}\|\_{L^2}$. It follows that
$$\|Lv\|=\|L(v-p)\| \leq \|L\|\|v-p\|\_{H^k} \leq C\|L\|\|v^{(k)}\|\_{L^2}.
$$


