I've been searching for a direct proof (I've since found a proof by contradiction in Royden's book) of the following problem:

Let $E$ be a measurable set and suppose $\{f_n\},f \in L^p(E)$ such that $f_n \rightharpoonup f$ i.e. $\{f_n\}$ converges weakly to $f$ in $L^p(E)$. Then there exists $M\in \mathbb{R}$ such that $||f_n||_p \leq M$ for all $n$.

By the Riesz Representation theorem, the weak convergence condition is equivalent to

$$\lim_{n \rightarrow \infty} \int_E f_ng = \int_Efg~~~ \forall g \in L^q(E)$$ where $q$ is such that $1/p + 1/q=1$.

I've been trying to leverage Holder's inequality with appropriate choice of $g$, but this hasn't done much, and I don't think will ever get me uniform boundedness, and this is the part I'm struggling with. How do I show there is $one$ $M$ such that $||f_n||_p \leq M$ for all $n$? Are there some often used techniques for doing this?



Yes, there is a general principle that is underlying this result, and it is called the uniform boundedness principle, which can be used to show that in every Banach space (not only $L_p$), a weakly convergent sequence is norm-bounded. See here.


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