# Math Help - precompactness in L p and C[0,1]

1. ## precompactness in L p and C[0,1]

Hi all,
I'm trying to learn precompactness arguments , I have a question about precompactness in function spaces $L_{p}$ and C[0,1] :
Prove the set $A = (sin(nt) )_{n=0}^{\infty}$ is precompact in $L_{p}(0,1)$ for $1 \leq p < \infty$ , but it's not precompact in C[0,1]. I looked at the theorems about equicontinuity and similar stuff but I couldn't find anything useful , can anyone give a hint??

2. Hi,
if I denote by $f_n$ the function $t\mapsto \sin (nt)$ we see that $f_n\left(\frac{\pi}n\right) = \sin \pi =0$ and $f_n\left(\frac{\pi}{2n}\right) = \sin \frac{\pi}2 =1$ so the $\delta$ in the definition of uniform continuity has to be smaller than $\frac{\pi}{2n}$. Hence we can not have the same $\delta$ for the whole family. $A$ is not equi-continuous so $A$ is not precompact.
To show the precompactness in $L^p$, I think you can use a criterion of equi-continuity in the mean for $L^p$.