Tensor Spaces: Part I

Before we can understand Tensor Spaces, we will need to understand vector spaces. As such, we’ll begin with inescapable definitions of mathematics preliminaries. If you have taken abstract algebra before, feel free to skip past these.

Field

Definition: An (algebraic) field (K, +, ·), is a set K with two maps

+ : K × K → K
 · : K × K → K

such that ∀ u, v, w ∈ K

For + map,
1. Commutative: u + v = v + u 
2. Associative: u + ( v + w) = (u + v) + w
3. Neutral Element: ∃ 0 ∈ K such that 0 + u = u + 0 = u
4. Inverse Element: ∀ u ∈ K , ∃ û ∈ K such that û + u = 0

For · map,
1. Commutative: u · v = v · u 
2. Associative: u · ( v · w) = (u · v) · w
3. Neutral Element: ∃ 1∈ K \ {0} such that 1 · u = u · 1 = u
4. Inverse Element: ∀ u ∈ K \ {0} , ∃ û ∈ K \ {0} such that û · u = 1

For + and · map combination,
1. Distributive: u · ( v + w) = u · v + u · w

Examples:

  • (ℝ, +, ·), that is usual operations of addition and multiplication on real numbers.

Terminology: An element of vector space is informally referred to as a vector.

Ring

A ring is a weaker notion of fields such that the · map is neither required to be commutative nor is there any need for existence of inverse element. Thus all fields are rings, but not all rings are fields.

Example: (ℤ, +, ·), that is usual operations of addition and multiplication on set of integers. Note that multiplication is commutative in this example, but not necessary.

K-Vector Space

Definition: A K-vector space (V, ⊕, *, K(+, ·)) is a mathematical structure with a set V and field K along with following two maps

⊕: V × V → V
*: K × V → V

such that ∀ u, v, w ∈ V and λ₁, λ₂ ∈ K, we have

For ⊕ map,
 1. Commutative: u + v = v + u 
 2. Associative: u + ( v + w) = (u + v) + w
 3. Neutral Element: ∃ 0 ∈ K such that 0 + u = u + 0 = u
 4. Inverse Element: ∀ u ∈ K , ∃ û ∈ K such that û + u = 0

For * and ⊕ map,
 1. Associative: (λ₁ · λ₂)* u = λ₁ * (λ₂* u) 
 2. Distributive I: (λ₁ + λ₂) * u = (λ₁ * u) ⊕ (λ₂* u)
 3. Distributive II: λ₁ * (u ⊕ v) = (λ₁ * u) ⊕ (λ₁ * u)
 4. Unity: ∃ 1 ∈ K such that 1*u = u

Examples:

  • A set P := {p: (-1, 1) → ℝ | p(x) = ∑ⁿ pᵢxⁱ } is not a vector space because no maps are specified.
  • However, (P, +, *, ℝ) is a vector space if
     + : P × P → P such that (p + q)(x) = p(x) + q(x) 
     * : ℝ × P → P such that (λ * p)(x) = λ · p(x)

Vector Subspace

Definition: U ⊆ V is a vector subspace for K-Vector space V if 
1. Closure under addition: ∀ u, v ∈ U ⇒ u ⊕ v ∈ U
2. Closure under s-multiplication: ∀ λ ∈ K, v ∈ U ⇒ λ * v ∈ U

Remark: We do not explicitly differentiate between ⊕ and + (or * and · ) because it is usually clear from context which operation is to be used.

Homomorphisms on Vector Spaces

Once we have space (informally a set with some structure), it is common to study in the maps between instances of such space using structure preserving maps called Homomorphisms. For vector spaces, such structure preserving maps are Linear maps.

Definition: Linear map f between vector space V ≡ (V, ⊕, *, K(+, ·)) and 
W ≡ (W, +̂, *̂, K(+, ·)) is f: V → W such that ∀ u, v ∈ V and λ ∈ K,

  1. f(u ⊕ v) = f(u) +̂ f(v)
  2. f(λ * u) = λ *̂ f(u)

A bijective linear map is called the vector space isomorphism. If vector space V is isomorphic to vector space W, we write V =̃ W.

Hom(V, W)

Hom(V, W) is a set all linear maps between V and W, that is
 Hom(V, W) := { f: V →̃ W} 
where “→̃” denotes a linear map.

Construction on Hom(V,W)

One can define a vector space (H(V, W), +, ·, (K, +, ·)) with maps
+ : Hom(V, W) × Hom(V, W) → Hom(V, W)
 · : K × Hom(V, W) → Hom(V, W)

such that for f, g ∈ Hom(V, W)
(f + g)(v) ↦ f(v) + g(v)
f(λ · v) ↦ λ cf(v)

Notice that the + and · can be different for V, W and K in general. Thus, Hom(V, W) is a vector space by itself.

Remark:
1. Endomorphism is Homomorphism of vector space on itself, 
 End(V) := Hom(V, V)
2. Automorphism is endomorphism that is also isomorphic, 
 Aut(V) := Iso(V, V)
3. End(V) and Aut(V) are themselves vector spaces.
4. Aut(V) ⊆ End(V)

Dual Vector Space

Dual vector space V* on vector space V is defined as Hom(V, K) where K is the underlying field on V considered as a field. 
Obviously, V* inherits the + and ·

In next part, I will begin talking about Tensor Spaces.