Should we glue it with Finer topology?
Tosha 15:08, 22 Feb 2004 (UTC)
- I don't think so. Weaker topology isn't the same as weak topology, which is often used with the specific meaning given in the current article. Lupin 23:26, 22 Feb 2004 (UTC)
Weak star topology
The current definition of the weak star topology is wrong. If is a reflexive space, then is an isometric isomorphism an the initial topology with respect to it therefore produces the norm topology on .
I know the terminology "weak-*-topology" only in cases where is a priori a function space and so the topology of pointwise convergence is meant. If there really is a general definition of "weak star topology" for non-reflexive spaces, then someone should write it in this section. If there isn't and all we have is the case of function spaces, then the section should be replaced by a diskussion of pointwise convergence (meaning: delete the false definition and only use the following section). 188.8.131.52 (talk) 19:33, 21 November 2010 (UTC)
Weak topology vs. initial topology
The first paragraph defines the weak topology as
- In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest or coarsest) topology on the set which makes all the functions continuous.
Is this a mistake or do some people really refer to the initial topology as the weak topology ? In particular strong topology, the opposite of weak topology, is not equivalent to final topology which makes this usage somewhat strange for me. MathMartin 13:03, 30 Apr 2005 (UTC)
- The term weak topology is certainly used in this sense. But there's no need to duplicate the definition of initial topology in this article, so I've modified the introduction to direct readers to the initial topology article instead. --Zundark 08:37, 30 May 2005 (UTC)
If the weak topology and the initial topology are the same thing, the correct solution is not to delete the redundant information from this article in favor of making the reader follow a link to read another article about the same thing. The correct solution is to merge the two articles. For now, I think the article suffers for clarity by not mentioning what exactly the weak topology "does". A brief definition of terms which may not be known is usual, even if that information is duplicated in that term's article. -Lethe | Talk 19:23, May 30, 2005 (UTC)
- Actually, I see now that the information about what the weak topology does is still made clear in the second paragraph. I retract my complaint about the removal of that information. We don't need it twice in the same article, right? -Lethe | Talk 19:26, May 30, 2005 (UTC)
- Yes, there's no reason to have it twice. Merging the articles is not a good idea, since they are about different things: the weak topology article is about the weak topology of a normed vector space and the weak* topology of its dual, while the initial topology article is about the general concept of initial topologies. --Zundark 20:02, 30 May 2005 (UTC)
The term weak topology is used in a wider context than that of functional analysis to mean the initial topology. I think the correct thing to do is to redirect weak topology to initial topology (since the terms are supposed to be synonymous) and rename this page to weak topology (functional analysis) or something similiar. -- Fropuff 15:22, 2005 May 31 (UTC)
- But the functional analysis meaning is the usual one, so redirecting it to some other article doesn't make much sense. --Zundark 18:41, 31 May 2005 (UTC)
Depends on who you are; to a topologist the topological meaning is probably the usual one. When there is more than one context for a specific title, it seems appropriate to link to the most general context applicable. We can put a note at the top of the initial topology page pointing here for purposes of disambiguation. -- Fropuff 18:55, 2005 May 31 (UTC)
I think there is some confusion. The strong topology as used in functional analysis is not a final topology. It is my understanding that the strong topology on a normed vector space X (or locally convex space) is the strongest topology on X to make a set of functions X→R (a linear subspace of the algebraic dual) continuous. Whereas the final topology on X is the finest topology to make a set of functions into X continuous. Thus although weak topology is an example of initial topology, strong topology is not an example of final topology.
The article weak topology is ok for the moment and should remain at this level of abstraction, as many people (e.g. physicists) need only this watered down version, so it deserves its own article. I propose the following renaming
- weak topology -> weak topology (normed vector space) (example of weak topology)
- weak topology (polar topology) -> weak topology (example of initial topology)
Perhaps weak topology (polar topology) should not be renamed and we could put a disambiguation article at weak topology. Similar considerations apply to strong topology. MathMartin 19:49, 31 May 2005 (UTC)
- Some topology textbooks (such as the book by Willard) use strong topology to mean final topology. In functional analysis, as you say, the meaning is different. Perhaps both, weak topology and strong topology should be disambiguation pages. -- Fropuff 20:30, 2005 May 31 (UTC)
What I'm about to say has been alluded to many times above, but with insufficient force. In topology and analysis, strong and weak do not merely mean different things, but opposite things, and the article should really reflect that. —Preceding unsigned comment added by 184.108.40.206 (talk) 05:08, 14 December 2007 (UTC)
- This may only serve to complicate matters, but the strong operator topology is actually an example of an initial topology, which some call a weak topology. Given the variety of meanings "weak topology" can take, I'd be in favour of disambiguating it, as with strong topology. It would also make lesser know weak topologies, such as the ultraweak topology easier to find. James pic (talk) 15:05, 17 December 2007 (UTC)