Date: Mar 18, 2013 3:00 PM Author: Kaba 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

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.

--

http://kaba.hilvi.org