# weak-* topology

Let $X$ be a locally convex topological vector space (over $\u2102$ or $\mathbb{R}$), and let ${X}^{*}$ be the set of continuous^{} linear functionals^{} on $X$ (the continuous dual of $X$).
If $f\in {X}^{*}$ then let ${p}_{f}$ denote the seminorm^{} ${p}_{f}(x)=|f(x)|$, and let ${p}_{x}(f)$ denote the seminorm ${p}_{x}(f)=|f(x)|$.
Obviously any normed space is a locally convex topological vector space so $X$ could be a normed space.

###### Definition.

The topology on $X$ defined by the seminorms $\{{p}_{f}\mid f\in {X}^{*}\}$ is called the weak topology and the topology on ${X}^{*}$ defined by the seminorms $\{{p}_{x}\mid x\in X\}$ is called the weak-$\mathrm{*}$ topology.

The weak topology on $X$ is usually denoted by $\sigma (X,{X}^{*})$ and the weak-$*$ topology on ${X}^{*}$ is usually denoted by $\sigma ({X}^{*},X)$. Another common notation is $(X,wk)$ and $({X}^{*},wk-*)$

Topology defined on a space $Y$ by seminorms ${p}_{\iota}$, $\iota \in I$ means that we take the sets $$ for all $\iota \in I$ and $\u03f5>0$ as a subbase for the topology (that is finite intersections^{} of such sets form the basis).

The most striking result about weak-$*$ topology is the Alaoglu’s theorem which asserts that for $X$ being a normed space, a closed ball (in the operator norm^{}) of ${X}^{*}$ is weak-$*$ compact. There is no similar^{} result for the weak topology on $X$, unless $X$ is a reflexive space.

Note that ${X}^{*}$ is sometimes used for the algebraic dual of a space and ${X}^{\prime}$ is used for the continuous dual. In functional analysis^{} ${X}^{*}$ always means the continuous dual and hence the term weak-$\mathrm{*}$ topology.

## References

- 1 John B. Conway. , Springer-Verlag, New York, New York, 1990.

Title | weak-* topology |
---|---|

Canonical name | WeakTopology |

Date of creation | 2013-03-22 15:07:05 |

Last modified on | 2013-03-22 15:07:05 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 46A03 |

Synonym | weak-* topology |

Synonym | weak-$*$ topology |

Synonym | weak-star topology |

Related topic | WeakHomotopyAdditionLemma |

Defines | weak topology |