Date: Mar 18, 2013 3:00 PM
Subject: Re: Maths pedagaogy
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
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
> 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
>> 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.