Wrongly closed question needs a better answer.

Question

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)...

Answer 1

For future reference, questions which are "based on an invalid premise" are valid questions for being closed under "unclear what you're asking", so this is not a wrong closure, procedurally speaking.

Setting aside the philosophical debate of what "clear/unclear what you're asking" means, this is a practical standpoint. It is often the case that both OP and answerers get lost in an XY problem or other issues on a question with invalid premises. It is rarely the case where such questions are of value for the community, and often the case where they are annoyances for people involved.

Whether they are properly described by "unclear what you're asking" may be debatable, but I think it is not controversial to say that this is the closest of our boilerplate close reasons which describe that phenomenom (being based on invalid premises).

Of course, that does not mean that there are no exceptions and that you cannot vote to reopen such questions which you think may have value.