Weak Convergence

Weak convergence is usually either denoted x_nw; ->x or x_n->x. A sequence {x_n} of vectors in an inner product space E is called weakly convergent to a vector in E if

 <x_n,y>-><x,y>  as n->infty,  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., ||x_n-x||->0 as n->infty. Schwarz's inequality now gives

 |<x_n-x,y>|<=||x_n-x||||y||  as n->infty.

The definition of weak convergence is therefore satisfied.

See also

Inner Product Space, Schwarz's Inequality, Strong Convergence

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Cite this as:

Weisstein, Eric W. "Weak Convergence." From MathWorld--A Wolfram Web Resource.

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