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 $\textbf{every}$ sequence $(y_n)$ made up of convex combinations of the $x_n$'s converges strongly to $x$?
In particular, 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
If $x_i= u_{floor(log(i)+1)}$ where $\{u_i| i=1,2,\ldots\}$ is an orthonormal basis for a Hilbert Space, then $(x_i)$ converges weakly to $0$ and $y_n = 1/n \sum_{i=1}^n x_i$ does not converge strongly to anything.
In particular, if $n = floor( e^k )$ for an integer $k\geq1$, then $||y_n - u_k|| < 2/3$ hence $||y_n||>1/3$.
Interestingly, if we define the linear operator $L_n: l_2\rightarrow R$ by
$L_n(x) = 1/n \sum_{i=1}^n x(i)$
then the $L_n$ converge arbitrarily slowly to the zero operator.