Let-a-gt-0-A-f-C-2-0-a-R-f-0-f-0-0-N-1-f-sup-f-x-f-y-x-y-0-a-N-2-f-sup-f-x-f-x-x-0-a-Prove-that-N-1-and-N-2-are-equivalents-norms-

Question Number 124415 by snipers237 last updated on 03/Dec/20

L e t a > 0, A = {f \in C^{2} ([0, a], R), f (0) = f^{'} (0) = 0}

N_{1} (f) = s u p {∣ f (x) ∣ + ∣ f^{″} (y) ∣, x, y \in [0, a]}

N_{2} (f) = s u p {∣ f (x) + f^{″} (x) ∣, x \in [0, a]}

P r o v e t h a t N_{1} a n d N_{2} a r e e q u i v a l e n t s n o r m s

Commented by mnjuly1970 last updated on 03/Dec/20

R e a l a n a l y s i s \dots