weak convergence

Suppose $X$ is a topological vector space, $X^{\prime }$ is the continuous dual of $X$, and $x_0,x_1,\mathrm{\dots }$ is a sequence in $X$. Then we say that $x_i$ converges weakly to $x\in X$ if

$$\underset{i\to \mathrm{\infty }}{lim}f(x_i)=f(x)$$

for every $f\in X^{\prime }$. The notation for this is $x_i\stackrel{𝑤}{\to }x$.

Title	weak convergence
Canonical name	WeakConvergence
Date of creation	2013-03-22 15:00:58
Last modified on	2013-03-22 15:00:58
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	8
Author	matte (1858)
Entry type	Definition
Classification	msc 46-00
Related topic	WeakConvergenceInNormedLinearSpace
Related topic	ConvergenceInDistribution