# Talk:Reflexive space

non-reflexive spaces: c0, l1, l_/infty ??

How can Reflixive space be defined as "Banach space", when below one can read "Montel spaces are reflexive." No infinite dimensional Montel space is a Banach space. —Preceding unsigned comment added by 83.23.122.42 (talk) 12:44, 22 March 2009 (UTC)

The 'Montel spaces are reflexive' statement is using the definition of reflexive for locally convex TVSs, while the lead only mentions the more restrictive case of reflexivity of normed spaces. Do you think this needs to be clarified? Algebraist 19:37, 22 March 2009 (UTC)
I agree with your change to the lede; for the definitions, I believe that the normed case is by far the most important (the only that was mentioned for a long time) so I feel that the normed case definition should be given before, although it is a special case of the TVS definition. --Bdmy (talk) 20:27, 22 March 2009 (UTC)

I am a bit dubious about the name "Kakutani Theorem". I have never heard of it, I cannot find any reference to it, and the mentioned result in Conway's book does not mention the name Kakutani. I suggest that the name be associated with a reference or dropped. Delio.mugnolo (talk) 23:02, 4 December 2011 (UTC)

## July 2013

If there are no objections, I'll separate the material devoted to the Banach (reflexive) spaces from the one about (reflexive) locally convex spaces, because what is written (in literature) about reflexive Banach spaces is disproportionally large (with respect to the rest), so that I believe that majority of readers prefer to get information about Banach spaces without going into the general locally convex situation. Besides this I want to create a separate article about super-reflexive Banach spaces and to remove there the material about them. Eozhik (talk) 09:45, 24 July 2013 (UTC)

A question to the authors: the notation ${\displaystyle J:X\to X''}$, where is it from? Is there a standard notation for this map? Eozhik (talk) 17:10, 24 July 2013 (UTC)

I can't force myself to make corrections that I want to make, because I suspect that I will damage the philosophical idea that the authors had in mind when writing this. Please, let me know would you mind if I define reflexive Banach space without references to the article about dual (topological vector) space, but instead on the base of the notion of dual normed space and dual norm? I've just made correction in the article on this topic. My idea is that it is easier to define reflexive Banach space without topology. We must use this possibility, because in my opinion, if something can be explained in a more simple way, the one who explains must use this way. What do you think about this? Eozhik (talk) 20:01, 24 July 2013 (UTC)

I am not sure that I understand what you mean by:
"I define reflexive Banach space without references to the article about dual (topological vector) space, but instead on the base of the notion of dual normed space and dual norm"
Is'nt it the way it was done already, by saying that the map J is onto? (by the way, I don't know where this notation comes from; Megginson uses Q, which I don't consider to be more standard than J) Bdmy (talk) 20:39, 24 July 2013 (UTC)

@Bdmy: Yes, but with one difference. I want to do the following:

Suppose ${\displaystyle X}$ is a normed vector space over a number field ${\displaystyle \mathbb {F} }$ (${\displaystyle \mathbb {F} =}$${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {F} =}$${\displaystyle \mathbb {C} }$) with a norm ${\displaystyle \|\cdot \|}$. Consisder its dual normed space ${\displaystyle X'}$, i.e. the space of all continuous linear functionals from ${\displaystyle X}$ into the base field ${\displaystyle {\mathbb {F} }}$ (${\displaystyle X'}$ is equipped with the dual norm ${\displaystyle \|\cdot \|'}$):
${\displaystyle f\in X'\quad \Longleftrightarrow \quad f:X\to {\mathbb {F} }}$
(${\displaystyle f}$ is a continuous linear functional).
This is a normed space (a Banach space to be precise), and we can consider its dual normed space ${\displaystyle X''=(X')'}$, i.e. the space of all continuous linear functionals from ${\displaystyle X'}$ into the base field ${\displaystyle {\mathbb {F} }}$ (${\displaystyle X''}$ is equipped with the norm ${\displaystyle \|\cdot \|''}$ dual to ${\displaystyle \|\cdot \|'}$):
${\displaystyle h\in X''\quad \Longleftrightarrow \quad h:X'\to {\mathbb {F} }}$
(${\displaystyle h}$ is a continuous linear functional).
Each vector ${\displaystyle x\in X}$ generates a map ${\displaystyle J(x):X'\to {\mathbb {F} }}$ by the following formula
${\displaystyle J(x)(f)=f(x),\qquad f\in X'.}$
It is easy to see that this is a continuoius linear functional on ${\displaystyle X'}$, i.e. ${\displaystyle J(x)\in X''}$. We obtain a map
${\displaystyle J:X\to X''.}$
Again it is easy to see that this map is linear. From the Hahn-Banach theorem it follows that ${\displaystyle J}$ is always injective and preserves norms:
${\displaystyle \forall x\in X\qquad \|J(x)\|''=\|x\|.}$
(i.e. ${\displaystyle J}$ maps ${\displaystyle X}$ isometrically onto its image ${\displaystyle J(X)}$ in ${\displaystyle X''}$). But ${\displaystyle J}$ may be not surjective.
If the map ${\displaystyle J:X\to X''}$ is surjective, then the normed space ${\displaystyle X}$ is called reflexive. This implies that ${\displaystyle X}$ is a Banach space (since ${\displaystyle X}$ is isometric to the Banach space ${\displaystyle X''}$).

The difference with what is written now is that in what I suggest there is no reference to the notion of dual topological vector space. In my opinion, this reference complicates the understanding of the question in Banach situation. To understand the idea here the reader must not know what topological vector spaces are, how one can define topology on the dual space, etc. So what I suggest makes the material easier. And this allows to separate the Banach situation from the locally convex one. (Of course, for the locally convex spaces one needs mentioning topologies on dual spaces). What do you think? Eozhik (talk) 05:53, 25 July 2013 (UTC)

I have no objection to this, I had the feeling that it was precisely done that way, I never took seriously the link to dual topological vector space. I think that what you say is the "normal" treatment, as appears in Banach space books, and also for example in the WP article Banach space. Bdmy (talk) 07:43, 25 July 2013 (UTC)
OK. In this case, I'll make these corrections. What I described will be the main, conceptual changes, but I think there will be some others, less essential, so I hope you will look and share your opinion. One technical question: is there a canonical notation for this map, ${\displaystyle J:X\to X''}$? Would you mind if I change it? Eozhik (talk) 08:13, 25 July 2013 (UTC)

Bdmy, I now think that maybe you were right when telling that there is no need of removing the material about super-reflexive spaces to a new article, but just in case, what were your reasons? Eozhik (talk) 06:22, 25 July 2013 (UTC)

Well, I have no objection to move the most technical content of the section "super-reflexive spaces" to a new article, especially if that section is going to grow more. I just wanted to say that I would like to keep a short summary of the "super-reflexive" notion in the "Reflexive space" article, in case the creation of a separate article is done, with a link to {{ main article }} as usual in WP. Bdmy (talk) 07:43, 25 July 2013 (UTC)
Also, Dear Eozhik, please take into account my request that you fill some "edit summaries" for your edits! These summaries are helpful for the people who have the article on their "Watchlist" and who help prevent WP from being damaged by vandals or bad editors. Bdmy (talk) 07:43, 25 July 2013 (UTC)
"...fill some "edit summaries" for your edits" -- excuse me, that was my fault. This is because I rarely come to WP. Yes, I also foresee that the material about super-reflexive spaces will grow, so maybe this will be inevitable to create a new article and to move there the main material (of course with keeping a summary here and with links). Maybe you should think about doing this yourself, since you are the author? To tell the truth, I would hesitate to do this, since I am not a specialist in this field. Eozhik (talk) 08:09, 25 July 2013 (UTC)
We'll see for a creation. So far I was too lazy for this! I'll be away for a few hours, hope something remains of this "Reflexive space" article when I'll be back!!! About the notation J, I don't care so much, but you'll have to have the support of a good book to choose a different one, otherwise there is no reason to change. I agree that Kolmogorov and Fomin is a good book, but too much of it will make this look as being ru.wikipedia!!! (just joking...) Bdmy (talk) 08:59, 25 July 2013 (UTC)
I guarantee that something will remain... But you should be prepared for the worse... :) Eozhik (talk) 09:06, 25 July 2013 (UTC)
Bdmy, I think, it's enough for today. Eozhik (talk) 12:08, 25 July 2013 (UTC)

### Footnotes

Bdmy, I want to ask you, how do you make footnotes? When I am clicking at "James (1972)" in the list of notes, the article by James in the list of references brightens. How do you do this? And another question: is it possible to rename the WP article devoted to the strong topology on dual space? Now it is called Strong topology (polar topology), and this sounds strange. I think it must be renamed to "strong dual space" or something like this. Eozhik (talk) 08:14, 26 July 2013 (UTC)

I haven't seen your post so far. For the first question, I saw that you found answers by yourself. Often, you can manage by looking at the "source code" that others have generated. You can also learn a lot on the help pages, for example if you type "Template:harvtxt" in the "search box", or many other "Template:so and so". I have so far no feeling for renaming the article you mention. I suppose that this should not be done without a very serious reason, as it creates linking problems and also globally affects WP I suppose. At least the change should be discussed on the corresponding "talk page" and perhaps at Wikipedia talk:WikiProject Mathematics (a place where I don't use to post myself). Bdmy (talk) 11:01, 26 July 2013 (UTC)
This is strange, now my footnotes look like yours... I didn't understand how this happened... OK.
But as to the Strong topology (polar topology) my reproaches to the author of this article are the following:
1) it is strange that the name of the object in the title contains parentheses,
2) in particular, that is why it is difficult to find this term if you need it: if you put "strong topology" into the search box, you'll be ejected to this article: Strong topology,
3) the very term "polar topology" is not used (I found it only in the book by Robertsons), it sounds strange and intuitively it means nothing (since each locally convex topology can be treated as the one generated by polars to some system of sets in the dual space).
As to links, I thought there must be a procedure of checking and correcting the links when something is renamed. Does something like this exist? Anyway I'll put this suggestion in the talk page of this article (or, maybe here: Wikipedia talk:WikiProject Mathematics), so thank you! Eozhik (talk) 12:56, 26 July 2013 (UTC)
For footnotes: I checked the syntax of "harvtxt". You can use
{{harvtxt|Schaefer|1966|loc= put what you want here}}
The first parameters are very strict (the "year" stands by itself as one parameter), in order to find the target. I don't know why, but targets introduced by "cite book" do not work for what I want, but "citation" does.
I do not feel terribly concerned about the article Strong topology (polar topology), and I am affraid that I will leave the problem to you, sorry! However: It is not rare at all that titles have parentheses, in order to disambiguate (for example Martingale (probability theory)). It is also normal to be sent to a disambiguation page like Strong topology, since some of the uses of the word, like "strong operator topology", must send to a different article. Finally, this discussion is a bit off topic here.
I am not expert about general procedures at WP. So far I only work locally on specific articles. Bdmy (talk) 15:06, 26 July 2013 (UTC)
"I checked the syntax of "harvtxt". You can use..." -- I see that you already corrected this, thank you! As to strong topology, you are right, this is not a good place to discuss this. Eozhik (talk) 16:40, 26 July 2013 (UTC)