18.3.2013 8:21, quasi 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:) 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 style-wise (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 >> 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".
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 subspace-based terms more convenient.