$$f \in C[0,1], f > 0$$ $$\lim_{p\to\infty}(\int_0^1f ^p(x)dx)^\frac{1}{p}$$ It is easy to see that the sequence is bounded above by the $\sup f$, but how can I bound it from below to prove that the limit is equal to $sup f$?
Asked
Active
Viewed 39 times
0
-
1It is bounded below by $0$, since $f>0$. – Clayton Apr 17 '18 at 21:57
-
So you only need a lower bound? – Apr 17 '18 at 21:58
-
if you know limit exists, then it has a lower bound – Apr 17 '18 at 21:59
-
I need bound it by the sequence converging to $sup f$ to prove that limit is $sup f$ – Ilya Fedorov Apr 17 '18 at 22:01