Consider the sequence of functions $(f_n)$ defined in the following way: $$ f_n(x) = \begin{cases} 0 & \text{ if } x \in [\frac{i}{n} + \frac{i}{n^2}, \frac{i+1/2}{n} - \frac{i}{n^2}] \\ 0 & \text{ if } x \in [\frac{i+1/2}{n}, \frac{i+1}{n} - \frac{i}{n^2}] \\ 1 & \text{ if } x = \frac{i+1/2}{n} - \frac{i}{n^2} \\ 1 & \text{ if } x =\frac{i+1/2}{n} +\frac{i}{n^2} \end{cases} \quad \text{ for } i = 0,1, 2,3,..., n-1 $$ and interpolating with a straight line between 0 and 1 in the missing intervals (this looks like triangular spikes that eventually that get more and more dense).
- Is this sequence relatively compact in $L^p([0,1])$?