Math 6210 • Real Analysis

\begin{enumerate}

\item
Give the definition of what it means for a sequence \((x_{n})\) in a Banach
space \(X\) to converge weakly to \(x \in X\).


\item
Let \(X = [0, 1]\) equipped with the Lebesgue measure. For \(n =\)
1, 2, \ldots, let \(f_{n} = n \cdot \chi_{[0,1/n]}\), where \(\chi_{A}\) denotes the indicator
function for the set \(A\). Prove that the sequence \(f_{n}\) does not converge
weakly to 0 in \(L^{1}([0, 1])\).


\end{enumerate}