This question was closed (I was one of those voting to close) and now has two votes to reopen. I didn't leave an explanation when I voted to close; I admit that I was rushed, it felt a clear-cut case to me, and I wasn't the last or the first to vote so thought I could get away without leaving a reason. As it now has votes to reopen, let me briefly explain my reasoning:

1. The level is inappropriate. The best answer to this question is "Read an introductory book on Functional Analysis". To understand a more subtle answer than "The HBT allows us to do FA" (pun ruefully acknowledged but not intended), the OP would have to have a good grounding in functional analysis obtained by reading such a book. There is no evidence from the question that the OP has done so. Moreover, if someone asked a question along the same lines but who had the appropriate background, I believe that they would have asked a very different question (but which would have elicited the same answers, and more).

2. The OP is not interested in the answer. This particular person has asked a stream of like questions and shows very little interest in the actual answers. There is no evidence that this user has looked for similar questions (such as my question on a space that needs the HBT).

3. Good answers do not make good questions. Indeed, "good answers" to poor questions are not good answers, they are good blog posts or wiki pages. Most of the answers given would be better placed on the wikipedia or nlab page about the HBT where they will be more easily found. I know that one of the intended side-effects of MO is to build up a repository of information. However, it is dangerous to make this the primary goal. In particular, one should think of the layers that one has to go through to dig out information buried in answers.

I can easily think of a better question on HBT that would elicit at least the same answers as this one. If the current question were replaced by such, I would happily vote to reopen. I leave it as a challenge to those voting to reopen to rephrase it into such a question.

(I just edited the title of this discussion. I copied the title from the question in question and didn't think about the fact that the "[closed]" might imply that this discussion had been closed!)

(I also forgot to link back to the original question: http://mathoverflow.net/questions/26568/why-is-the-hahn-banach-thorem-so-important-closed.)

@Harry: We have learned, haven't we, not to speculate on hidden motives and behavior, especially negatively? I see no evidence that this user is not reading the responses to his/her questions. A reasonable assumption of good faith (okay, a core policy of a different website, but a good one) would suggest that someone who repeatedly asks questions which display some familiarity with high level mathematics on a Q&A website like ours would read the answers s/he gets, otherwise why continue to ask questions?

Nevertheless, I agree with your core point: this user's recent practices have been somewhat exasperating, the more so because most of his/her questions are eliciting responses from expert mathematicians (including me).

I don't necessarily think that refusing to reopen this question is the answer (although I will not vote to reopen it) but rather that veteran users who follow these threads should be aware that this user's behavior is somewhat problematic and take this into account when they decide whether to spend time crafting an answer to a vague question.

So it seems the problem here is that this was asked by a problem user and so people paying enough attention to notice that closed it, but people who only saw this one question disagree.

As to the particular question, I think it's of interest to people who aren't analysts themselves but interact with analysists enough to want to understand what they're talking about. For example, this describes of me. I haven't taken an analysis class in 8 or so years and have no recollection of what HB says or why it's useful. On the other hand, I do research related to operator algebras and go to OA conferences. And they're always using these basic functional analysis results and I'd love to know a few paragraphs about why they're interesting. (I suppose I should just read a functional analysis text, but the fact of the matter is I'm not going to, but I would read some MO answers.)

Okay, delete the 8 words "has asked a stream of like questions and" from my original post. The rest still stands and is sufficient, in my opinion, to justify voting to close the question.

@Andrew: I disagree that this question should be closed. Here are my replies to your numbered reasons (it's late, so I won't write much more than that):

1. Sure, HBT allows us to do FA, but I think it's valuable to have concrete examples that would make it obvious how crucial it is. Just pointing to a Functional Analysis book is useless -- a person who doesn't already know any FA won't know what page to look at, and even if he did, won't know what many of the words mean. (For example, if you open Rudin's book on FA, you'll find that every proof of every fact refers to a million facts from previous chapters, making it impossible to read out of order).

2. I have no idea how well-meaning the questioner is, but this has no effect on how good the question is. It's fine if he never looks at it again as long as other people do and learn something from it.

3. I simply disagree (although I understand that quite a few people here do agree with you). I'm quite convinced that good answers do make good questions. I'm too tired to write up reasons but hey, there's a badge for that!

Finally, I agree with what you said about improving the question. If closing it were the best way to make people rewrite it, I'd be all for it. But first, I think you should've left a prominent comment there saying this was your intention. And secondly, I doubt it will get rewritten (unless you do it; the original poster might simply be unable to do it well), and even if it is, what will happen to the answers already there?