Talk:Weak topology

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Quick question: is there any reason for the author(s) of the subsection "Weak-* convergence" to switch back and forth between \phi and \varphi? — Preceding unsigned comment added by 2001:638:502:A006:213:72FF:FE9F:2852 (talk) 11:25, 9 May 2016 (UTC)[reply]

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 $X$ is a reflexive space, then $T:X\to X^{**}$ is an isometric isomorphism an the initial topology with respect to it therefore produces the norm topology on $X$ .

I know the terminology "weak-*-topology" only in cases where $X$ 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). 78.53.90.64 (talk) 19:33, 21 November 2010 (UTC)[reply]

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)[reply]

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)[reply]

They're not really different things though, are they? The same thing in two different contexts, one general, one specific. -Lethe | Talk 18:19, May 31, 2005 (UTC)

Yes, but product topology and subspace topology are also specific examples of initial topology. Merging them all into a single article wouldn't be very helpful. --Zundark 18:41, 31 May 2005 (UTC)[reply]

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)[reply]

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.

"is the strongest topology on X to make a set of functions X→R... continuous."

Wouldn't that be the discrete topology? Silly rabbit 21:13, 12 June 2006 (UTC)[reply]

Weak topology (on a normed vector space) is just a simple example of the more general concept of weak topology (polar topology) on locally convex vector spaces (or dual pairs).

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)[reply]

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 72.221.121.69 (talk) 05:08, 14 December 2007 (UTC)[reply]

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)[reply]

Given the variety of meanings and the potential for confusion I think a disambiguation page would really be best here. -- Fropuff (talk) 16:40, 17 December 2007 (UTC)[reply]

Bounded weak topology

Bounded weak topology is described in German wikipedia: [1]. Boris Tsirelson (talk) 21:01, 4 April 2015 (UTC)[reply]

On the use of the term "strong topology"

This article calls "strong topology" to what I think it should be called "norm topology", since "strong topology" has a different meaning for spaces of operators between normed spaces, which are of course normed spaces themselves. — Preceding unsigned comment added by 157.92.4.4 (talk) 17:20, 29 May 2015 (UTC)[reply]

Exceptionally bad article

There is a false belief about Wikipedia mathematics articles, namely that as long as they are not mathematically incorrect, that is the only criterion for being "good".

No. It is essential that they explain their subject matter the way you would explain it to a friend who wants to learn about the subject.

Any article that delves into the mathematical formalism of the weak topology without stating what the open sets of this topology are, or at least a base for the open sets, or at the very least a subbase for the topology, is a truly bad article.

I have trouble understanding how anyone could imagine writing such an article without about a topology without stating at least a subbase for the open sets.

Not far, far down in the article, but right where the topology is first described. 2601:200:C000:1A0:55C0:140D:2395:B94 (talk) 23:08, 9 April 2021 (UTC)[reply]

About the "topology of uniform convergence"

I consider that this expression is misleading, since for the topology derived from the usual norm of continuous linear maps between normed spaces, convergence does not mean uniform convergence on the whole space but only on the unit ball - because in the formula for the norm, the supremum is taken on this ball. May-be one could accept this expression due to the fact that uniform convergence on the whole space would mean for linear maps the same as with discrete topology. But a warning seems to be needed. Moreover this still could create confusion in contexts where also non linear maps are discussed ... UKe-CH (talk) 10:25, 31 March 2023 (UTC)[reply]

"Weak compactness" listed at Redirects for discussion

The redirect Weak compactness has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 October 29 § Weak compactness until a consensus is reached. 1234qwer 1234qwer 4 20:51, 29 October 2023 (UTC)[reply]

Theorem from Rudin wrongly quoted

The quoted theorem says 3.10 Theorem Suppose X is a vector space and X' is a separating vector space of linear functionals on X. Then the X'-topology tau' makes X into a locally convex space whose dual space is X'.

The statement in this article is: If Y is a vector space of linear functionals on X, then the continuous dual of X with respect to the topology σ(X,Y) is precisely equal to Y.(Rudin 1991, Theorem 3.10)

Problems: So first of all, it is missing the assumption that Y is separating points which is important for this theorem to hold.

Secondly it seems like the tau' topology is actually the weak topology on X' and not the weak topology on X. 134.2.85.160 (talk) 08:21, 30 July 2024 (UTC)[reply]