Weak limit of disjoint normalized sequence in $L^p$

Question

I want to prove that the weak limit of a disjoint normalized (pairwise disjoint supports, elements of norm $1$) sequence $(f_n)$ in $L^p$ for $p >1$ is zero ?

I started with the measure of the support of each function $f_n$ , but is it true that it goes to zero ?

Answer 1

Let $S_j$ be the support of $f_j$, $S:=\bigcup_jS_j$ and $q$ the conjugate exponent of $p$ (that is $p^{-1}+q^{-1}=1$). Then for each $g\in L^q$ and each $n_0$, we have $$\int f_n\cdot \operatorname{sgn}(f)|g|\chi_{S_{n_0}}\to \int |f|\cdot|g|\chi_{n_0}$$ as $n\to +\infty$. But the LHS is $0$ when $n>n_0$ so for each $n_0$, $f\chi_{S_{n_0}}=0$ hence $f\chi_S=0$.

As $\{f_n\chi_S\}=\{f_n\}$, this sequence converges weakly to both $f$ and $f\chi_S=0$, and a weak limit is unique, so we can conclude.

Answer 2

One way to prove it is by using the following 2 facts 1- a normalized disjoint sequence in $Lp$ is basic, and 2- every basic sequence in reflexive space $X$ is weakly null !