Weak Convergence

Weak convergence is usually either denoted $x_n{w\atop\rightarrow} x$ or $x_n\rightharpoonup x$. A Sequence $\{x_n\}$ of Vectors in an Inner Product Space $E$ is called weakly convergent to a Vector in $E$ if

$$\left\langle{x_n, y}\right\rangle{} \rightarrow \left\langle... ...uad \hbox{as }n\rightarrow\infty,\quad {\rm for\ all\ }y\in E.$$

Every Strongly Convergent sequence is also weakly convergent (but the opposite does not usually hold). This can be seen as follows. Consider the sequence $\{x_n\}$ that converges strongly to $x$, i.e., $\Vert x_n-x\Vert\rightarrow 0$ as $n\rightarrow \infty$. Schwarz's Inequality now gives

$$\vert\left\langle{x_n-x,y}\right\rangle{}\vert\leq \Vert x_n-x\Vert\, \Vert y\Vert\quad\hbox{as }n\rightarrow\infty.$$

The definition of weak convergence is therefore satisfied.

See also Inner Product Space, Schwarz's Inequality, Strong Convergence