The question I am asked is:
Find a bound sequence in $l^p$, $L^p([0,1])$ and $C([0,1])$ respectively, which has no convergent subsequence.
I guess I have found such sequences in $l^p$ (namely $(x_n)_n$ with $\xi_{n,k}=1$ if $n=k$ and $\xi_{n,k}=0$ if $n\ne k$) and $C([0,1])$ ($x_n(t)=t^n$) but I am not able to find such a sequence for $L^p([0,1])$. Since there is no measure given, I am assuming I should prove it for the Lebesgue-measure. If it were the counting measure, for example, it would be easy.
If I want to find such a sequence with the Lebesgue-measure I can not let the norm of the functions of my sequence diverge to infinity. I also can not find a good way of dividing my interval $[0,1]$ into (countably) infinite subintervals which all do not have measure $0$ and do not converge.
Thank you in advance!