# weak convergence

Suppose $X$ is a topological vector space, $X^{\prime}$ is the continuous dual of $X$, and $x_{0},x_{1},\ldots$ is a sequence in $X$. Then we say that $x_{i}$ converges weakly to $x\in X$ if

 $\lim_{i\to\infty}f(x_{i})=f(x)$

for every $f\in X^{\prime}$. The notation for this is $x_{i}\xrightarrow[]{w}x$.

Title weak convergence WeakConvergence 2013-03-22 15:00:58 2013-03-22 15:00:58 matte (1858) matte (1858) 8 matte (1858) Definition msc 46-00 WeakConvergenceInNormedLinearSpace ConvergenceInDistribution