Consider the following theorem. Let $X$ be a Banach space, and suppose $x_n$ is a sequence in $X$ converging weakly to $x$. Then $\|x\| \leq \liminf \|x_n\|$.
Clearly this resembles Fatou's lemma. Is there a concise way to express Fatou's lemma as the above theorem?
After a little bit of thought, there are some subtleties involved. Let $f_n(x)$ be a sequence of nonnegative measurable functions on a measure space. Then define $f(x) = \liminf f_n(x)$. These functions and their $L^1$ norms could be infinite, so some of the standard Banach space theorems need to be modified to correctly interpret Fatou's lemma which states $\int f d\mu \leq \liminf \int f_n d\mu$.