Linguistic Note
11
Tensor Product
TANAKA Akio
1
Field F
Linear space V, W
Additive group X
Map f : V × W → X
F-bilinear map satisfies below condition.
α1, α2 ∈V β∈W
f ( α1 + α2, β) = f (α1, β ) + f (α2, β )
α∈V β1 , β2∈W
f ( α, β1 + β2) = f (αβ1) + f (αβ 2)
α∈V β∈W λ∈ F
f (λα, β) = f (α ,λβ )
2
Field F
Linear space V, W
Additive group T
F-bilinear map τ∈ BL ( V ×W, T )
F-bilinear map f ∈ BL (V×W, X )
Additive group’s homomorphism f ~ : T → X
Tensor product of V and W satisfies below condition.
Pair ( T, τ)
f = f ~ . τ
3
Tensor product V ( T, τ)
Tensor product W ( T `, τ` )
Additive isomorphism φ : T ≅ T `
τ` = τ . φ
4
Tensor product is expressed by the following briefly.
V ⊗ W
5
Algebra on field F A, B
Tensor product A ⊗ B is defined by the following.
x, y ∈ A ⊗ B
xy : = ( mA ⊗ mB ) ( h ( x ⊗ y ) )
h : = ( A ⊗ B ) ⊗ ( A ⊗ B ) ≅ ( A ⊗ A ) ( B ⊗ B )
mA ⊗ mB : ( A ⊗ A ) ⊗ ( B ⊗ B ) → A ⊗ B
6
Algebra on field F A, B, C
f ∈ Hom Fal ( A, C )
g ∈ Hom Fal ( B, C )
α ∈ A β∈ B
f (α) g(β) = g (β) f (α)
h (α ⊗ β ) = f (α) g (β)
[Note]
Relationship between bilinear map τ , τ ` and isomorphism φ , namely τ` = τ . φ, may be helpful to word ( τ ) , sentence ( τ` ) and grammar ( φ ).
[Reference]
Frame-Quantum Theory Tokyo March 13, 2005
Frame-Quantum Theory Addendum Tokyo March 26, 2005
Frame-Quantum Theory map 3 Tokyo Hakuba March 28, 2005
Compendium Premise for Frame-Quantum Theory Tokyo March 22 – April 10, 2005
Tokyo July 30 2007
Sekinan Research Field of Language
