Linguistic Note 11 Tensor Product / July 30 2007

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.

α_1, 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 : = ( m_A  ⊗ m_B ) ( h ( x  ⊗ y ) )

h : = ( A ⊗ B ) ⊗ ( A ⊗ B ) ≅ ( A ⊗ A ) ( B ⊗ B )

m_A  ⊗ m_B : ( A  ⊗ A ) ⊗ ( B  ⊗ B ) → A  ⊗ B

6

Algebra on field F      A, B, C

f ∈ Hom _F^al ( A, C )

g ∈ Hom _F^al ( 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