Consider the Mazur's Lemma (H. Brezis – "Functional analysis, …"):
Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that converges strongly to $x$.
The Lemma says that "there exists a sequence…''.
Is it true that every sequence $(y_n)$ made up of convex combinations of the $x_n$'s converges strongly to $x$?
For example, if we consider
$$y_n = \frac{1}{n}(x_1 + x_2 + … +x_n),$$
is it true that $y_n$ converges strongly to $x$?
Best Answer
Short answer: $y_n=x_n$ is a convex combination of $x_1,\dots,x_n$. Clearly, this need not converge strongly.
Longer answer: for any particular choice of coefficients one can cook up a sequence $(x_n)$ where the strong convergence of convex combinations fails. Here's an idea. For any sequence $a_n\to \infty$ we have $\sin a_n t\to 0$ weakly in $L^2[0,1]$. Choose $a_n$ such that they grow slowly, for example $a_n=\lfloor \log_4 n\rfloor$. You will observe that when $n = 4^{k+1}-1$ for an integer $k$, most of the terms $x_j(t)=\sin a_j t$ for $1\le j\le n$ are equal to $\sin k t$; about $3/4$ of them are. So, the average can be estimated pointwise from below: $$ |x_1+\dots+x_n|\ge \frac34n |\sin kt| - \frac14 n $$ hence $$ \frac1n |x_1+\dots+x_n|\ge \frac34 |\sin kt| - \frac14 $$ This shows that $|y_n|\ge 1/4$ on the set where $|\sin kt|\ge 2/3$, and this set has measure that doesn't get small as $k\to\infty$.