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• CommentAuthorAlex Bartel
• CommentTimeAug 24th 2011 edited

I hadn't seen the question that Will mentioned or the comments, but it seems rather extreme to "never use MO again" because of some sceptical comments on the first question and of a quote from her website, a link to which she provided herself. If, as others have said, this person had observed MO for a few days, she would have seen that the long time users do an important and thorough job, keeping the site useful for mathematicians by closing off-topic questions. To expect that at the same time, one's own questions will never come under any scrutiny is like saying I will never fly again, because the first time I flew, they checked me and my hand luggage, as though they assumed I was a terrorist. One doesn't have to think too many steps ahead to realise that this is done for one's own good (provided one really is a legitimate user).

I found the perspective of a new user as sketched by Thierry very valuable and interesting. But I agree with Will that for a mathematician it is not all that difficult to lurk for a few days or weeks and to do one's homework before participating, and to get a very warm welcome and to enjoy a smooth arrival.

@quid: Experience suggests that starting anonymously, one runs a much much higher risk of a bad start than by investing one's identity into the contributions.

• CommentAuthorgrp
• CommentTimeAug 24th 2011

We can't please or appeal to everyone. The situation Will mentioned was quite unfortunate, and should probably forgotten and put behind one. I will take away from it that there is great potential for misunderstanding, and that with new users who may bring benefits to the site, I should appear as much like a helpful and unassuming assistant as possible. I would attempt the care and consideration that Will showed, but I would also act in a way to allow the user to provide additional information at their desire.

This is with the intent of handling new questions that suggest mathematical maturity. Of course, if all new users read the FAQ and other helpful documents before acting, we would not have the number of discussions on meta that we have had.

1.
I think I tried to make too many points in my previous answer. One of the points that Alex and Will appear to be overlooking, and a point that I thought was important, is that **even** for someone who does their homework and waits a while to get a feel for the site before posting (and, of course, starts by posting answers before questions), the first few days of activities can be a bit nerve-wracking. And no amount of anecdotal evidence ("I played by the rules and I had a good experience") can change the fact that some people that should have stayed have left after a single bad experience early on.

MO does a good job of saying upfront that questions are taken very seriously and that people should think before asking. When you're a regular it goes without saying, it's no big deal, and we wouldn't want it any other way; but it puts a lot of pressure on the newbies. Again, I find the MO culture perfectly fine, I just think that we (including myself, btw) could do with a little bit more mindfulness when new users are involved.
• CommentAuthorquid
• CommentTimeAug 25th 2011

Alex, I do not think that one can infer from the, also IMO true observation, that among the new users one observes on the site anonymous ones have a much higher risk for a bad start that for one specific individual anon or not makes a large difference. In particular, if the new user pays some attention to the fact of being anon and therefore is moreorless forced to write some motivation (cf. my response to Tom Leinster above). In my opnion, but I have no data, a key point for a good first question is that the motivation is clear, and not in the mathematical sense, but really in the sense why did this user ask the question.

2.

quid, I agree with your assessment that the motivation is often crucial for making a question suitable for MO or interesting for others. On the other hand, you explicitly suggest that one reason for a new user for being anonymous would be the possibility to just start over again unscathed if the first contributions go wrong. In my experience, this kind of thinking does not tend to improve the quality of the contributions.

@Thierry Actually, you made that point perfectly clear, and I found it very interesting, I probably just failed to address it properly.

Clearly, some people left who should have stayed, but in some (although not all) cases it seems an overreaction to me, and I don't see anything specific in the MO culture that would have to change for particularly faint-hearted people not to leave (except to allow everything and to never question the poster's background or motivation, which would of course mean the doom of MO).

• CommentAuthorNilima
• CommentTimeAug 25th 2011

@Alex, I cannot speak for anyone else.

However: I perceive a difference between an atmosphere of professional questioning/confrontation and a personal one. Admittedly, sometimes this is subtle. For instance:

'This approach you suggest is completely wrong' is qualitatively different from 'You don't know anything about this'. Or:

'You have provided insufficient motivation and background for this problem, and we don't know at what level you seek the answer'
is different from
'I don't know who you are or why you are asking this' or 'I know you have expressed weird ideas elsewhere so why should I take you seriously now?'

The point is: at work, I try to err on the side of questioning, critiquing and attacking the message, and not the messenger. Sometimes I lapse. I will never be accused of being soft mathematically, but I try to be kind in person.

Of course one should seek clarification about a question if it is unclear. I think this can be done politely, professionally, and carefully, and many users already do this. Asking for clarification carefully doesn't mean everything is allowed. Walking out because one perceives an environment as unfriendly isn't faint-heartedness, it's honest.
• CommentAuthorAlex Bartel
• CommentTimeAug 25th 2011 edited

@Nilima Of course, faint-heartedness is honest. Madelina perceived the environment as unfriendly and walked out, and that seems to be largely due to Will's comments. As I said, I haven't actually seen the comments in the thread that Will mentioned. But I have never seen Will attack anyone personally on this site, nor have I ever seen him say anything that would have made me leave the site forever, even if he had said that to me when I was a newcomer. Moreover, it seems clear that Will had the best intentions. That's why I used the word faint-hearted. This is of course a subjective term and is meant to implicitly contain a comparison with how I would have reacted. I simply haven't seen a comment of Will's that would have justified walking out (in my personal view), and I doubt that the ones under discussion were a big exception from the rule.

Now, if I were to leave a comment to the effect of "don't be fooled by the superficial appearance of the question, note that the OP has a good mathematical background, here are some quotes from her home page" and that would prompt the person to walk out, the incident would leave me puzzled as to what I was supposed to do differently. That's the point I was trying to get across.

• CommentAuthorWill Jagy
• CommentTimeAug 25th 2011

Thank you, Alex. I do try, and this episode was upsetting to me as well.
• CommentAuthorNilima
• CommentTimeAug 25th 2011

Alex, I concur- in whatever I've seen, Will has always been both gracious and professional.

Thanks also for clarifying what you meant. Your comment was meant in a narrower sense than I realized.

In general, 'faint-hearted' has pejorative connotations, so perhaps we can collegially agree to an adjective less charged. How about 'daunted', or 'uncomfortable'?
3.

Alex, I concur- in whatever I've seen, Will has always been both gracious and professional.

I agree with this statement.

I know that when one says something that inadvertently upsets someone else, then if one is denied the opportunity to set the record straight it can leave a bad taste. Those of us (perhaps myself, I'm not sure) who tend to leave curt comments can learn from this that if someone can take even one of Will's courteous remarks the wrong way, ours are probably more susceptible to misunderstanding.

Over on TeX-SX we have a list of "templates" for likely messages for new users. The point of these isn't that they be prescriptive ("you must choose one of these") but that they lay down a minimum level of politeness. Even if one doesn't use one of these messages, the fact that they are there makes one think a little about the message one is about to leave. Thinking about it, I probably am more polite on TeX-SX than here. That's something I should probably fix.

• CommentAuthorgilkalai
• CommentTimeAug 26th 2011 edited

I also like Will's style and contributions overall. But, in my opinion, Will's first comment above regarding Madalina's question was insulting. (beside being irrelevant to this discussion, and incorrect.) If a mathematician is driven away from MO by insulting/patronizing/hostile reactions to a question he or she poses, we cannot merely regard it as a "misunderstanding". Perhaps we have to doublecheck ourselves.
• CommentAuthorquid
• CommentTimeAug 26th 2011

It seems to me that the general subject of communication on MO, in particular regarding those involving new users, can be an important discussion. However, discussing it along the lines of one particular incident (in particular this one) seems, for various reasons unfortunate to me (including, but not limited to, the fact that this thread is the meta thread of a particular question yet not the one discussed now; the actual incident is not known and is not knowable anymore to everybody, and what is still visible is so incomplete that it is misleading and, in a negative way, misrepresents Will Jagy's contribution).

http://tea.mathoverflow.net/discussion/1124/communication-on-mo-in-particular-with-new-users/ for those wishing

for those wishing to continue the general discussion, independent of the specific incident.

• CommentAuthorgilkalai
• CommentTimeAug 28th 2011

Regarding the original question: It looks that Andrew Stacy had a lot more to say about the mathematics and could have proivided a great MO answer that would have enlightend the rest of us. But instead Andrew (along with others) wanted to discover very fine details on the background of the person who asked the question and spent a lot of time and energy writing comments about this issue instead. I must say I don't understand it at all.
• CommentAuthorquid
• CommentTimeAug 29th 2011

Gil, it seems to me he [Andrew] explained in quite some detail why he does not/cannot answer the question [73246], for example he says "it is possible to read it in too many different ways and each has a subtly different answer." Do you want him to write an answer for each interpretation? (Leaving aside the fact that he, in this discussion and frequently before, expressed his believe that MO is not the place for lengthy and general expositions.)

What I do not understand is why the questioner or those who think the question is 'good' do not edit the question (or ask another question); as several people suggested quite some time ago. Or, at least explain in detail what they mean, or where they disagree with the reasoning of those who think otherwise.

To repeat and rephrase what I said in an earlier comment, my personal opinion is: The original question (as written) was unclear/vague. Ryan and Donu correctly guessed the intent (as confirmed by latter comments of the questioner). So, the original question (in its spirit) was answered in the comments (the first ones!).

• CommentAuthorgilkalai
• CommentTimeAug 29th 2011 edited

Quid, the original question was fairly clear and the motivation and background of the OP also became clear rather quickly. There was no need to further edit the question. There were two satisfying answers and then Andrew cryptically expressed, over several comments, his mathematical concerns with the answers and his ideas about what a good answer should be. Some elaboration on this mathematical issues in an answer could have been much more fruitful than repeatedly nagging the OP. If there are some subtle issues that the OP and rest of us are not aware of then it can be very good to say what they are. (This is quite common for various MO questions.) This could be beneficial to the OP and also (which is not less important) to other mathematicians (including those who offered earlier answers) who found the mathematical issue interesting.
4.

@Gil: The question wasn't clear enough for many people. I think when you get questions that are sufficiently outside the scope of MO's mandate you invariably get confusion. The question has several flaws that have been pointed out already, and the OP appears to have abandoned it so it's not clear to me why the question was re-opened, especially without editing.

• CommentAuthorquid
• CommentTimeAug 29th 2011

Gil, let us simplify this discussion: In the second comment of SPG to SPG's answer two different interpretations are given. Which one is the one that was (originally) intended by the questioner?

• CommentAuthorgilkalai
• CommentTimeAug 29th 2011

(Based on the question and his comment) the OP asked the following " I work as a postdoc at the moment, but vector bundles and topology aren't my area. I've realised that they might be very useful for something I'm doing at the moment. So I've got Milnor and Stasheff's book and I'm missing some motivation. I would like to understand why (for the purposes of the theory developed in the book) we define infinite dimensional R(inf) the way it is defined in the book, and especially why we consider sequences with only finitely many non zero entries.

This question is very clear. It may or may not be more suitable to the sister site but there is no more information or clarifications the OP can give us, or any editing which will make the situation clearer.

SPG's answer in the comment "because its technically convenient" is not entirely satisfying (as the answer "since the authors are famous mathematicians we can trust their definitions") The answer "It's the appropriate limiting object, and that's the role it's meant to play -- to be something that contains all the finite-dimensional Euclidean spaces, but 'no more' " gives a clear impression that Ryan is, as usual, on top of things but that he does not really make the effort to explain matters to the OP(which is fine). SPG's answer is not that different than the comment: the definition is technically convenient and if you replace direct sum with direct product, God knows what happend. Allen's answer point out that this definition ensures that the Grasmanian with this definition is a CW complex which allows to use tools from algebraic topology and Allen also points out that in some cases when you consider "too large" objects things may unexpectingly become homotopically trivial.

Andrew, in a series of comments criticizes the current answers and hints that there are several models to be chosen, all homotopically equivalent, each has its advantages and disadvantages; that the situation is actually mathematically quite interesting and he can tell us much more about it except he won't, at least not before knowing what is the OP's shoe size and how tall he is (or something like that). Only then, we will be able to know if the question deserves the good answer Andrew can give (and I am sure he can), or perhaps the answer is more suitable to a wiki that will never be written and that we will never see even if it will be written. As I said, I dont understand this approach.
• CommentAuthorspg
• CommentTimeAug 29th 2011

Gil,

I agree with you that my answer is not entirely satisfying. As I commented in reply to Andrew, the OP asks:

"My question is why do we insist that only finitely many of the xi are non-zero for each (x1,x2,x3,…)∈R∞? "

This is the only sentence in the OP's post ending in a question mark. I think that this is unambiguous without any need for further clarification or interpretation. As for whether the question is suitable for MO, from the FAQ

"MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books."

Again, it is not really debatable: one comes across this question when reading Milnor+Stasheff, a graduate level book. It's an obvious question, and given current practices maybe better suited for MathSE than MO. Perhaps the problem is that the FAQ needs to be changed to better reflect current practices of 3000+ point users?

Quid, my comment was meant as rhetorical, I do not really believe there are 2 interpretations of the question.
• CommentAuthorquid
• CommentTimeAug 29th 2011 edited

Gil, yes this comment seemed clear to me (and further clarified an initially vague or if you prefer fairly clear question). As such it might have been useful to edit this information into the question rather than to only have it as a comment in a fairly long comment thread. Perhaps there was some need for editing after all. Yet, in my opinion this question got already answered within the first three comments, before that comment was even made. So, I did not understand the continuing discussion. Except if some people still read it differently than I read it, which in turn would imply the situation is still not clear. Or, the request is for elaboration on the comments, but I honestly never understood the discussion to be mainly about elaboration on these comments, except until your last comment. [Added: where by 'the discussion' I mean your comments and the ones by Andrew you refer to as well as my contribtions; of course earlier parts where in some sense about elaboration or 'level'.]

SPG, thank you for the information. But this starts to be a bit confusing for me. The following questions seem quite different to me:

a. What is the/a motivation of M&S to use the definition they use [direct sum]?

b. Could one use this specific alternative definition [direct product] instead and still do what M&S do?

c. Could one use some other alternative definition to do what M&S do, or does one have to use the one they use?

d. Is the specific alternative definition used/usefull anywhere else in this context?

e. Are other alternative definition used/usefull anywhere else in this context?

All of which I could imagine, in principle and abstractly [I do not have the expertise to truly judge this], to be things on could answer 'around' this question. I believe that the main intent was to ask a., in particular based on the comment of the OP quoted by Gil Kalai, I repeat the keypart "[...]I would like to understand why (for the purposes of the theory developed in the book) we define infinite dimensional R(inf) the way it is defined in the book[...]"; and before that suspected something like this based on the first reaction to comments. However, it seems to me you think b. is asked. While some of Andrew's comments perhaps suggest that also an answer to c. would be interesting. So, I am confused.

5.

I still believe the question is a bit too vague. But I decided to answer what I assume is the OP's main concerns, given that they're reading Milnor and Stasheff.

• CommentAuthorEmerton
• CommentTimeAug 29th 2011 edited

I agree with Gil and SPG's comments above, to the effect that the question was pretty unambiguous (why consider the direct sum rather than direct product of counably many copies of R), and that a context was provided (a non-expert reading Milnor and Stasheff), and that furthermore this context placed the question squarely in the (stated) purview of MO, namely a question come across while reading a graduate-level book.

I don't see why this question generated so much fuss.

I also wonder what the (actual, rather than stated) point of MO is at this stage: if someone can't come and ask a (possibly confused, but still essentially unambiguous) question about Milnor and Stasheff, what is the minimum technical level of question that people (say those participating in this thread) regard as appropriate?

• CommentAuthorYemon Choi
• CommentTimeAug 29th 2011 edited

Why not consider the direct sum?

Ill-fitting parable: when thinking about C*-algebras, you can consider the c_0-sum of a (countable) family of C*-algebras. Why is this defined the way it is?

• CommentAuthorYemon Choi
• CommentTimeAug 29th 2011

Put another way: I am sure there are theorems about the Grassman-ish object defined as the set of n-dimensional subspaces of $\prod_{n=1}^\infty {\mathbb R}$, equipped with suitable structure. I suspect they are not the same theorems as the ones about the Grassmanian of n-dimensional subspaces of ${\mathbb R}^\infty$. Without having a copy of Milnor-Stasheff to hand, and not being familiar with the book, it is not clear to me which of these theorems would be the ones of interest.

• CommentAuthorspg
• CommentTimeAug 29th 2011

Emerton,

I also find myself wondering what the actual purpose of MO is. This question seems an interesting test case.

On the one hand, I am somewhat sympathetic to those who believe that the question is ill-suited to MO. It has the "feel" of a MathSE question to me, in that it's not a question that would interest an expert in geometry very much---it's the sort of thing one might wonder about out of idle curiosity, or, as happened in this case, because one is exploring an alien land and hasn't figured out the customs yet.

On the other hand, the fact is that it was asked in good faith by a young professional mathematician trying to understand an area far from her or his own. In such situations, it should be easy for professionals in that area to clear up the confusion quickly, or else to reassure that it is something the experts don't know. Is this not what MO is for?

I am a professional mathematician who often wonders about such questions outside my own area of expertise. When I can't ask a local expert to help with a question I suspect should be routine, does this mean that I should turn to MathSE for help?

spg
6.

I think I can see both sides.

On the one hand, when you're having trouble understanding a new piece of mathematics, it can be hard to formulate a really focused question. You're fumbling around in the dark, and probably you don't know exactly what it is that's blocking your understanding. So the best you can do is "why is this defined the way it is?", and you hope that someone knows what you mean well enough that they can tap into the source of your confusion and enlighten you. I'm sure there have been questions like this on MO before, and everything's gone just fine.

On the other hand, there were some particularly unfortunate circumstances in this case. Several people genuinely found it hard to know what kind of answer the OP wanted. (The first person to say so on this thread was Qiaochu, who I have never seen being petty or anything other than level-headed.) Ordinarily that would be OK: if commenters request clarification, the OP generally clarifies. That's all part of the process. But in this case the OP didn't, and in fact reacted in a quite emotional way. If he/she had promptly edited the question, or even just written "sorry, I'm a beginner at this stuff and don't know how to make my question any more precise", that would probably have defused things.

I haven't communicated with Andrew about this, so the following is pure guesswork, but I wonder whether for him it was uncomfortably close to a question of the type "write me an expository article about such-and-such". Evidently he could think of lots of things to say on this topic, but he didn't know which ones would be useful to the OP. And when he asked, he didn't get a reply that helped him to narrow it down. So I think I can understand his frustration.

Later on, unpleasant things were written by two other anonymous users, on this forum and in a swiftly-deleted answer on the main site. Even if you think that some people were simply pretending not to understand the OP, there's no one to blame for those pieces of nastiness other than those who wrote them.

• CommentAuthorquid
• CommentTimeAug 29th 2011 edited

Emerton, this relevant context was provided after the closure, in fact I guess as a consequence of it; and it wasn't a quick closure, and the OP commented before it. Might I ask that Gil, you, and whoever else in addition wishes to tell the closers how wrong they were to at least acknowledge this fact.

SPG, a. or b. or is this the same?

• CommentAuthorspg
• CommentTimeAug 29th 2011

quid,

My apologies. I read your list too quickly. I suppose neither a) nor b), but rather exactly what I quoted the OP asking above (just to avoid confusion, I'd prefer not to endorse any paraphrasings of the OP's question). This was the original question and has not been edited: so I believe the OP's question was there, clearly stated, for all to see who read carefully. I cannot speak for Emerton, but I do not wish to tell the closers how wrong and evil they were, but merely that they erred if their reason for closing was that the question was ambiguous.

spg
• CommentAuthorquid
• CommentTimeAug 29th 2011

SPG, thank you for the response.

• CommentAuthorEmerton
• CommentTimeAug 29th 2011

Dear Quid,

I think "evil" may be a bit of an extreme adjective to introduce into the discussion; I haven't seen it used, or intimated, before now (unless I missed something).

I looked through the question again at the various timestamps, and saw that you are right and I was wrong vis a vis the non-expert/Milnor and Stasheff material. Nevertheless, the question of direct sum vs. direct product is expressed from the very beginning, and the additional material providing context was posted in under 24 hours. And although it wasn't originally made explicit, the fact remains that this question did arise in a legitimate way from reading graduate level texts. (And I find it hard to think that anyone would regard this question as dealing with undergraduate level material.)

In any event, it may be that such questions, asking for very basic clarifications of graduate-level subjects, no longer belong on MO, but then, as SGP suggested, perhaps the FAQ should be updated to reflect this.

Best wishes,

Matthew

7.

I'm sorry to interrupt this interesting discussion, but let me throw in a somewhat related "MO rule of thumb":

MO is a good place for mathematicians to ask basic questions in a field outside of their area of expertise.

The reason for this rule is mainly sociological, but it is nevertheless a viable rule. Many professional mathematicians would feel uncomfortable asking questions on MSE or similar sites since those sites are primarily intended for less a experienced audience. Indeed, the most suitable responses to such questions is most likely above the usual standards of these alternate sites. (Note that MSE often has very excellent answers to questions, so don't take this last sentence to mean that MSE is not a good place to ask such questions!)

I don't like throwing big names around just for show, but let me give this example. Last year, Terry Tao asked some relatively basic questions on ultrafilters. Granted that ultrafilters aren't a standard part of the graduate curriculum, but any expert on the topic will concur that these questions are basic knowledge for the area. Terry asked because these questions were relevant to his current work but outside his current knowledge base. Would anyone refer Terry to MSE or elsewhere in such circumstances? Of course not! Is Terry the only mathematician worthy of this exception? Of course not!!!

• CommentAuthorquid
• CommentTimeAug 29th 2011

Dear Emerton,

it seems 'evil' has a stronger meaning than I thought. I retract it and appologize. Thank you for confirming the time-line.

Regarding the general question whether typical graduate-level material should be on-topic on MO or not, I agree that a clarification and/or discussion could be useful; indeed, I made a somewhat similar observation a week ago in this thread. Personally, I would not have anything against MO being (or perhaps again being) more open towards this. However, it seems to me this particular question is not the best example for making a case for it.

Thanks again and best wishes!

8.

Quid gave five distinct interpretations of the question. From the question itself I cannot tell which the OP is asking and each, in my mind, requires a different answer. That is why I do not like this question. I think that the level is absolutely fine for MO and that any of those five questions would be perfectly acceptable here. But unless or until the OP clarifies which question they mean, then I cannot see how it is answerable.

My best guess is that the first is the right answer, and then Ryan's answer comes the closest to answering it; though I still think that he isn't clear enough. The key to answering the first version would be to explain exactly why this particular model is a good one to use. Allen says "Because then it is a CW-complex", but so what? Why is actually being a CW-complex better than having the homotopy type of a CW-complex (this would be a very good question, I think)? (Then he goes off with some irrelevance about U(∞) versus U(ℋ)). Ryan at least says:

A key nice result about the weak topology on ℝ<sup>∞</sup> is that any continuous function from a compact space to ℝ<sup>∞</sup> has an image in ℝ<sup>k</sup> for some k.

which to me, at least, is the heart of the matter. It says that when dealing with ℝ<sup>∞</sup> then you are effectively dealing with "very big (but finite) ℝ<sup>k</sup>", at least if your source space is compact (say, a closed manifold or finite CW-complex). So although we want to deal with the classifying space BU(n), we can pretend in any given circumstance that it is a finite dimensional manifold/CW-complex.

But the rest of Ryan's answer, and of what just about everyone else says, is pretty much model independent and so talks about properties of BU(n) without saying why one particular model is preferable to any other. If the OP is truly asking "why this model and not another" then the answer has to address some property of this model that is not held by another, and explain why that property is important.

In response to other remarks, no matter how often I read the question I do not see any mention of the direct product. There are a heck of a lot of spaces between the direct sum and the direct product which I would expect Jo Mathematician to be vaguely familiar with, far more familiar with than the direct product. With no information as to the field of the OP, I don't see how we can assume that he or she means to compare the direct sum with the direct product. (For what it's worth, the direct product is countable infinite dimensional: the direct sum is dense in it.)

This is the sort of question where even if I don't answer it myself, I feel I am competent enough to judge what is a good answer to it. There is not enough information in the question for me to be able to do that! Allen's answer I just do not like, Ryan's is okay, SPG's doesn't address the issue of different models, Paul Garrett's could be taken in one of two ways: either it is about the homotopy type (in which case it doesn't address the issue of different models) or it is about the specific model (in which case it doesn't address what is special about this particular model), Yemon's is - sadly - also missing the point: most of the time we only care about the homotopy type of BU(k) so the particular model doesn't matter, it would surprise me if M&S's book couldn't work with a different model.

To summarise: there is nothing in this question to indicate that it is of a level below that of MO. That part of the debate I find quite bizarre. However, there is also nothing in this question to indicate exactly what sort of answer would satisfy the OP. I was commenting on it because this is the sort of question where I might have been able to contribute, but without knowing more then I wouldn't know exactly what to contribute. That the OP was satisfied with Allen's and Donu's answers (though exactly how, I have no idea) means that the OP needn't bother responding to my comments. But I still maintain that it is not a good question, and we have not have any good answers yet (though I applaud those who tried for doing so).

• CommentAuthorquid
• CommentTimeAug 30th 2011 edited

Andrew, since I also mentioned 'level'. The question is not the problem, but what about this comment:

"[...] It seems to me that the right hand side of the equality is (possibly) "bigger" than the left hand side. Let $1^k \in \mathbb{R}^k$ denote the finite, constant sequence, $(1,\ldots,1)$. The limit of $1^k$ as $k$ tends towards infinity does not lie in $\mathbb{R}^{\infty}$, even though we can identify $1^k$ with $(1,\ldots,1,0,0,\ldots) \in \mathbb{R}^{\infty}$ for all $k < \infty.$ I can see that all of the elements of $\mathbb{R}^{\infty}$ can be constructed by the union, but we seem to be able to construct other elements too."

In my opinion, this confusion is the root of the question.

ADDED: I should say 'was' instead of 'is' as it got resolved. But, in my opinion, this 'answered' the original question.

9.

Yes, that comment did reveal considerable confusion on behalf of the questioner. Given that the questioner was confused about that, I'm surprised that the apparently satisfactory answers were satisfactory. But I may be being uncharitable, this might have just been one of those "hadn't thought it through" incidents.