Kaba wrote: >quasi wrote: >>Kaba wrote: >>> >>>From this our assertion is obvious. QED. >> >> I'll try to clarify ... >> >> By definition, g is non-degenerate on E if ker(g) = 0. > >--8x-- (full proof) > >> It follows that g is non-degenerate on E iff g is non-degenerate >> on each E_i. >> >> I agree that the proof could use more detail, but I also think >> that the student, at the level expected for learning the material >> in that textbook, should be able to supply those details. > >I think our proofs illustrate that Lang abuses the word obvious:)
Not really. I simply continued where Lang left off, applying the definitions.
>Both proofs are tremendous improvements over Lang's. I would >summarize this as: either provide the proof, or make it an >explicit exercise; but don't do both.
I disagree. It's perfectly OK for an author to develop a proof to the point where the rest is easy, and leave the remainder for the reader to think through.
>It is easy to become blind to such issues after you have already >mastered the subject.
Yes, but the best teachers and authors have mastered the art of judging how their presentation would be seen from the point of view of the prospective student.
>>> In addition to problems with clarity, Lang does not actually >>> use g anywhere; he uses a dot instead. Such problems show up >>> when you don't polish, i.e. read through, your writing >>> carefully. >> >> Such problems show up when the student jumps into a chapter >> without reading the author's stated notational conventions. >> >> At the beginning of the chapter, on the bottom of page 571, >> Lang clearly states that for notational convenience, if there >> is no potential ambiguity, he will use <v,w> or even v.w as a >> shorthand for g(v,w). > >That's true. However, this is still bad style-wise (which is the >main point of my replies). Such definitions should be local.
Lang specifies his choice of notation and terminology at the beginning of the chapter and uses it throughout the chapter.
Repeating those definitions would be silly.
>For example, you don't want to hunt for the domain of function >f from 8 pages back (been there).
That's not what we're talking about.
We're talking about a simple notation which he plans to use repeatedly. He defines it once at the start -- that's sufficient. But in any case, those definitions were on the page before the proof in question, not 8 pages back.
>>> Since I had hard time with Lang's proof, I ended making up my >>> own proof; that proof follows next. Perhaps these can be >>> compared for clarity. >>> >>> My proof >>> -------- >> >> Your theorem: >> >>> Let V be a bilinear space, where the bilinear form is either >>> symmetric, alternating, or hermitian. Let U, W subset V be >>> subspaces of V, such that V = U _|_ W. Then V is >>> non-degenerate if and only if U and W are non-degenerate. >> >> You're mangling Lang's terminology. >> >> In the context of the given chapter, Lang uses the terms >> "degenerate" and "non-degenerate" as properties of a given >> bilinear form, not as properties of a space. >> >> Some other authors apply those terms in the same way as you, >> but Lang does not. > >Page 573: "Instead of saying that a form is non-degenerate on E, >we shall sometimes say, by abuse of language, that E is >non-degenerate".
Right, I missed that.
But in trying to explain Lang's proof to you, I only needed to read page 571, which has the definitions, and page 572 which had the statement of the theorem, and the top of page 573 which had the proof. There was no need to read further.
But had I read just a little further (to the end of page 573), I would have seen that Lang does indeed introduce the alternate terminology you refer to above.
Still, if you are giving an alternate proof of Lang's theorem (Theorem 1.1), you should either state the theorem in the same way, or, if you choose to state the theorem differently, you should make it clear that your statement of the theorem is equivalent to the one in the text.
>Anyway, even if that didn't read there, since it's my theorem >and proof, I get to choose:)
Sure, but as I said above, if you choose a notational convention different from that of the theorem you asked about, you should say so up front.
>In practice, once you start working with bilinear spaces, you'll >quickly find the subspace-based terms more convenient.
Yes, I agree, provided only one bilinear form is being considered,