Mathematics GRE

http://ejde.math.txstate.edu/Volumes/2011/18/chen.pdf

this is articl; I ha ve question in page 8 proof of lemma (4.5) which is:
I want to know why he must indicate to weak star convergence(Alaoglu's Theorem)???
There he she wrote:' Since K(x)\in L^p(\omega),we have |u_m|_p,\gamma';

and thank you.

This is to distinguish it from the normed topology. A closed ball in the space is always compact in the weak* topology, while it is compact in the normed topology if and only if L^P(\Omega) is finite dimensional.