0

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?

Lilla
  • 2,099
  • 1
    The steps in https://math.stackexchange.com/a/4551755/115115 should transfer directly to the vector case. – Lutz Lehmann Oct 19 '22 at 20:02
  • @LutzLehmann This does indeed fully answer the question: you could make it either an answer, or we could mark my question as duplicate – Lilla Oct 19 '22 at 20:22

0 Answers0