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
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)
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,
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 sgnΣi<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.