# 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 ${x}_{i}\stackrel{\mathit{w}}{\to}x$.

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 |