Kaba
Posts:
289
Registered:
5/23/11


Re: Maths pedagaogy
Posted:
Mar 18, 2013 3:00 PM


18.3.2013 8:21, quasi wrote: >>From this our assertion is obvious. QED. > > I'll try to clarify ... > > By definition, g is nondegenerate on E if ker(g) = 0.
8x (full proof)
> It follows that g is nondegenerate on E iff g is nondegenerate > 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:) 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.
It is easy to become blind to such issues after you have already mastered the subject.
>> 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 stylewise (which is the main point of my replies). Such definitions should be local. For example, you don't want to hunt for the domain of function f from 8 pages back (been there).
>> 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 >> nondegenerate if and only if U and W are nondegenerate. > > You're mangling Lang's terminology. > > In the context of the given chapter, Lang uses the terms > "degenerate" and "nondegenerate" 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 nondegenerate on E, we shall sometimes say, by abuse of language, that E is nondegenerate".
Anyway, even if that didn't read there, since it's my theorem and proof, I get to choose:) In practice, once you start working with bilinear spaces, you'll quickly find the subspacebased terms more convenient.
 http://kaba.hilvi.org

