Let $f: (a,b) \rightarrow H$, for $H$ real Hilbert, belonging to $H^2(I,H)$.
Consider a uniform subdivision of $I$ of width $\Delta x$ and $\pi f$ the linear Lagrange interpolant of $f$:
is it true that $||f-\pi f ||\leq C \Delta x ^2 ||f||_{H^2(I,H)}$ ?
The answer is positive for $H$ finite dimensional: one can apply the Bramble-Hilbert lemma to all components and conclude. But what about an infinite dimensional vector space? Is there any additional condition to $H$ to make the result hold, or is a counter-example known?