weak convergenceWeakConvergence2005-02-07 13:49:412005-02-11 16:05:05Definition% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}Suppose $X$ is a topological vector space, $X'$ is the continuous dual
of $X$, and $x_0,x_1,\ldots $ is a sequence in $X$.
Then we say that $x_i$ \emph{converges weakly} to $x\in X$ if
$$
\lim_{i\to \infty} f(x_i) = f(x)
$$
for every $f\in X'$. The notation for this is
$x_i \xrightarrow[]{w} x$.