Deveno, In an effort to understand your language, I took some notes from Halmos, Finite Dimensional Vector Spaces, and pass them on as a lanquage excercise:

These are essentially lanquage notes, not math notes.

Linear funnctional on a vector space V is a scalar valued function y defined for every vector x with the property that, for scalars ai

y(a1x1+a2x2)=a1y(x1)+a2y(x2)

The set V’ of all linear functionals on V is a vector space (A set of elements is a vector space if…..). V’ is called the dual space of V.

Def. The annihilator S0 of any subset S of V is the set of all vectors y in V’ st [x,y], (y=f(x)), =0.

Def. Dual Basis: A basis of V’

Theorem: If X is an n-dim basis of V, there is an n-dim basis of V’.

Def. Direct Sum U(+)V of U and V (over F): All ordered pairs <x,y>, x in U and y in V.

Def: Tensor Product U(X)V is the dual of all bilinear forms on U(+)V.

Surprinsingly, we can piece it together.

Theorem: W=U(+)V is a vector space.

Def. w(x,y) is the value of w at <x,y>

Def. w is a bilinear form, (or functional) if:

w(a1x1+a2x2,y) = a1w(x1,y)+a2w(x2,y)

w(x,a1y1+a2y2)=a1w(x,y1)+a2w(x,y2)

If w(x,y)=a1w1(x,y) + a2w2(x,y), w is a bilinear form, the set of all bilinear forms onW is a vector space.

If z belongs to W, the set of all functions y(z) is the dual of W.

NEXT: Permutations, Multi-Linear forms, Alternating Forms, and Determinants.

Def: A permutation P maps S(n)onto S(n).For ex, if P(1)=2, P(2)=3, P(3)=1,

P(1,2,3)=(P(1),P(2),P(3))=(2,3,1)

P(3,1,2)=(3,1,2)

If Q and R are arbitrary permutations:

(QR)i=Q (Ri), E exists st EP=PE=1

P’ exists st P’P=PP’

The set of permutations form a group. (Remember, these are from Halmos’ notes, not my knowledge)

The representation of a permutation as a product of permutations is not unique.

At this point the development goes into the std def of sgnP (Mirsky,Perlis), as sgnΣ_{i<j}(xi-xj), but in such an obtuse development that I finally gave up. The only point being that any transposition changes the sign of P so that any sequence of even transpositions always has a + sign and any sequence of odd transpositions always have a - sign.

The whole permutation development was simply an excruciating expositon on the Levi-Civita symbol, eijk…

So I looked ahead to Alternating Forms, and determinants, and gave up. For a mathematician this may be mother’s milk; for my limited capacity, it’s like flying to Paris to potty and then not being able to find one.

I must admit, up to permutations it was a some-what interesting logical succession.

But it’s all yours Deveno.