# Wrongly closed question needs a better answer.

This question, more precisely this version, was closed on not-clear-what-you're-asking grounds.

That's simply wrong; it's perfectly clear what he's asking. He "defines" two Hilbert spaces, $$H_1=L^2$$ with the standard norm and $$H_2=L^2$$ with a different norm, and he wants to construct an isometry between the two.

It's clear what he's asking, the problem is that the second norm is simply not a norm on $$L^2$$, so the question is based on an invalid premise. So an answer pointing out the problem and then correcting the question and answering the corrected version seems appropriate.

There's an answer that was posted before the question was closed. The first version of that answer was wrong, purporting to answer the question as stated. I pointed this out in a comment, and the guy's willing to fix it. But his first revised answer is also wrong, for slightly more subtle reasons.

The point is that the second norm makes a different space into a Hilbert space. The correct definition of that different space involves some subtleties. Trying to get the existing answer fixed by the answerer editing in response to my comments is like pulling teeth - if I could simply post an answer it would work much better. It would be an answer that people who care about such things but don't understand them very well would find interesting and educational:

Sketch The point is that, speaking very informally, the second space should consist of the $$f$$ such that $$f,f'\in L^2$$. Here's where it gets interesting and subtle: What people typically mean when they say $$f,f'\in L^2$$ is not literally $$f,f'\in L^2$$. (The problem with the current version of the existing answer is that it takes $$f,f'\in L^2$$ literally, which simply doesn't work.)

In fact "$$f,f'\in L^2$$" is typically (and here should be) shorthand for any of a long list of equivalent conditions. Seems to me a proper answer really should include at least the statement of a long TFAE theorem - we're never going to get there via the present comment-edit-comment-edit cycle.

So I should vote to reopen? Did that. I gather there exist $$n$$ such that $$n-1$$ more reopen votes would do it (hint)...

• The question is now open. I am curious as to why you didn't post this question here. – Xander Henderson Aug 26 at 21:09
• To reiterate "Unclear" means, at least can mean, "unclear as written." It is not wrong to put on hold a question that is unclear as written. The very idea of 'on hold' is that OP, or anyone, can edit to make it clear as written. It can be alright to have a vagueish question when one searches for the correct notion. What is a problem though is when the question says that something impossible must be done. In a way that's a detail of formulation, yet it should be avoided to counter the problem of answers showing what was claimed because it "must" work. I'll edit the Q a bit. – quid Aug 26 at 21:37
• I cast the first vote to close as unclear only once OP had edited the question in response to comments with idle speculation about what the question might mean, which is a rather bizarre response given that they were the one who asked the question in the first place. That is, even after it had been pointed out the question had a false premise, they refused to clarify what they actually wanted the question to mean. That to me makes the question unclear. – Eric Wofsey Aug 26 at 21:51
• @XanderHenderson I wasn't aware of that thread, thanks. – David C. Ullrich Aug 26 at 22:46
• @EricWofsey I wasn't aware of that aspect of it - wan't paying enough attention I guess. – David C. Ullrich Aug 26 at 22:46
• @EricWofsey I wasn't aware that the OP had edited the question in an attempt to fix the problem with the original. Yes, the new version is much wackier than the original. I hope the question nonetheless remains open, because in my not very humble opinion my answer adds value to MSE. (Hmm, maybe I could find a better question to migrate it to???) – David C. Ullrich Aug 27 at 14:55
• @EricWofsey ("Comments may only be edited for five minutes...:) In particular it seems possible that one part of the theorem is not well known to people who know about Sobolev spaces; I've never seen it stated elsewhere... – David C. Ullrich Aug 27 at 15:04