Weak Sequential Convergence in $L^{p}$

Characterize the bounded linear functionals on Euclidean space $\mathbb {R}^{n}$. Show that for a sequence in $\mathbb {R}^{n}$, the concepts of strong convergence, weak convergence, and componentwise convergence are the same.

For $1 \leq p < \infty$, assume that both $\left\{ f_{n}\right\} \rightarrow h$ and $\left\{ f_{n}\right\} \rightarrow g$ weakly $L^{p}(E)$. Show that $g = h$.

For $1 \leq p \leq \infty$, and $q$ the conjugate of $p$, assume that $\left\{ f_{n}\right\} \rightarrow f$ weakly in $L^{p}(E)$ and $\left\{ g_{n}\right\} \rightarrow g$ strongly in $L^{q}(E)$. Show that

At what point does the proof of Theorem 14 fail in the case $p = 1$ and also in the case $p = \infty$.

Theorem 14: If $1 \leq p < \infty$, then every bounded sequence in $L^{p}(E)$ has a subsequence that converges weakly to a function in $L^{p}(E)$.

Show that if a sequence in $C[a, b]$ converges weakly then it converges pointwise. Find a bounded sequence $C[a, b]$ that fails to have any weakly convergent subsequence.

Verify the assertions made regarding the convergence of the Radamacher functions.

If $\left\{ f_{n}\right\} \rightarrow f$ weakly in $L^{\infty }[a, b]$, show that

Fix real numbers $\alpha$ and $\beta$. For each natural numbers $k$ and $n$, consider the step-function $f_{n}$ defined on $I = [0,1]$ by

Show that $\left\{ f_{n}\right\}$ converges weakly in $L^{2}(I)$ to the constant function that takes the value $(\alpha +\beta ) / 2$. For $\alpha \neq \beta$, show that no subsequence of $\left\{ f_{n}\right\}$ converges strongly in $L^{2}(I)$.

Show that $\left\{ f_{n}\right\} \rightarrow f$ strongly in $L^{2}(E)$ if and only $\left\{ f_{n}\right\} \rightarrow f$ weakly in $L^{2}(E)$ and $\left\{ \left\| f_{n}\right\|_{2}\right\} \rightarrow \left\| f\right\|_{2}$.

Use Problem 20 to show that, for $1 \leq p < \infty$, a bounded sequence in $\ell^{p}$ converges weakly if and only if it converges componentwise.