867:, 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 ...
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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
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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
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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
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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?
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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.
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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.
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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
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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. -
938:
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'.
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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
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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)
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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
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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
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There is a false belief about
Knowledge mathematics articles, namely that as long as they are not mathematically incorrect, that is the only criterion for being "good".
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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.
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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?
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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.
729:, 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
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should be replaced by a diskussion of pointwise convergence (meaning: delete the false definition and only use the following section).
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Problems: So first of all, it is missing the assumption that Y is separating points which is important for this theorem to hold.
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They're not really different things though, are they? The same thing in two different contexts, one general, one specific. -
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Yes, there's no reason to have it twice. Merging the articles is not a good idea, since they are about different things: the
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Given the variety of meanings and the potential for confusion I think a disambiguation page would really be best here. --
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But the functional analysis meaning is the usual one, so redirecting it to some other article doesn't make much sense. --
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is an isometric isomorphism an the initial topology with respect to it therefore produces the norm topology on
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Secondly it seems like the tau' topology is actually the weak topology on X' and not the weak topology on X.
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article is about the weak topology of a normed vector space and the weak* topology of its dual, while the
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Weak topology (on a normed vector space) is just a simple example of the more general concept of
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continuous. Thus although weak topology is an example of initial topology, strong topology is
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I think there is some confusion. The strong topology as used in functional analysis is not a
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Not far, far down in the article, but right where the topology is first described.
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Knowledge:Redirects for discussion/Log/2023 October 29 § Weak compactness
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article is about the general concept of initial topologies. --
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The current definition of the weak star topology is wrong. If
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is the strongest topology on X to make a set of functions X→R
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I know the terminology "weak-*-topology" only in cases where
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Some topology textbooks (such as the book by
Willard) use
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Bounded weak topology is described in German wikipedia:
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Is this a mistake or do some people really refer to the
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to determine whether its use and function meets the
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