weak convergence

Suppose $X$ is a topological vector space, $X^{\prime }$ is the continuous dualof $X$, 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$x_i\stackrel{𝑤}{\to }x$.