PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: Very high
weak-* topology (Definition)

Let $ X$ be a locally convex topological vector space (over $ \mathbb{C}$ 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) = \lvert f(x) \rvert$, and let $ p_x(f)$ denote the seminorm $ p_x(f) = \lvert f(x) \rvert$. Obviously any normed space is a locally convex topological vector space so $ X$ could be a normed space.

Definition 1   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-$ *$ 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 $ \{ y \in Y \mid p_\iota(y) < \epsilon \}$ for all $ \iota \in I$ and $ \epsilon > 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'$ is used for the continuous dual. In functional analysis $ X^*$ always means the continuous dual and hence the term weak-$ *$ topology.

Bibliography

1
John B. Conway. A Course in Functional Analysis, Springer-Verlag, New York, New York, 1990.



"weak-* topology" is owned by jirka.
(view preamble | get metadata)

View style:

See Also: weak homotopy addition lemma

Other names:  weak-* topology, weak-$*$ topology, weak-star topology
Also defines:  weak topology

Attachments:
weak-* topology of the space of Radon measures (Example) by stevecheng
Log in to rate this entry.
(view current ratings)

Cross-references: term, functional analysis, Reflexive, similar, compact, operator norm, closed ball, Alaoglu's theorem, basis, intersections, finite, topology, normed space, seminorm, continuous dual, linear functionals, continuous, locally convex topological vector space
There are 15 references to this entry.

This is version 5 of weak-* topology, born on 2005-03-08, modified 2005-03-09.
Object id is 6855, canonical name is WeakTopology.
Accessed 9477 times total.

Classification:
AMS MSC: 46A03 (Functional analysis :: Topological linear spaces and related structures :: General theory of locally convex spaces)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)