5.14.2 Tensor Canonical Map Target Space

Tensor Canonical Map Target Space is the vector space V ⊗ W viewed specifically in its role as the codomain of the canonical product map ⊗: V × W → V ⊗ W, rather than as an abstract algebraic object considered on its own. Examining V ⊗ W as a target space emphasizes the properties it must have to correctly receive the image of the bilinear pairing operation and to support the factorization of every other bilinear map defined on V × W.

Structural Requirements of the Target Space

A Vector Space Generated by Simple Tensors

As a target space, V ⊗ W must be a vector space in which the images v ⊗ w of the canonical map span the entire space. No proper subspace of V ⊗ W can contain the image of ⊗, since the universal property requires factorizing maps to be determined by their values on a spanning set.

No Relations Beyond Bilinearity

The target space must impose no relations among simple tensors other than those forced by bilinearity itself. If the target space identified two simple tensors that bilinearity did not require to be equal, some bilinear map would fail to factor consistently through it, violating the universal property.

Distinguishing the Target Space from the Domain

Multiplicative versus Additive Dimension

As previously noted for the domain product, the dimension of the target space is multiplicative rather than additive in finite dimensions:

This reflects that the target space must be large enough to encode all possible products of basis vectors from V and W as independent basis tensors.

Basis of the Target Space from Bases of the Factors

If {eᵢ} is a basis of V and {fⱼ} is a basis of W, the target space V ⊗ W has as its basis the collection of simple tensors {eᵢ ⊗ fⱼ}, one for each pair of basis indices. Every element of the target space is uniquely a linear combination of these basis tensors.

The Target Space as the Home of Factored Maps

Where the Unique Linear Map Lives

Given any bilinear map β: V × W → Z, the induced unique linear map f: V ⊗ W → Z has the target space V ⊗ W as its domain. In this sense, the target space of the canonical map doubles as the domain from which all bilinear information is subsequently processed linearly.

Minimality of the Target Space

Among all vector spaces that could serve as the codomain of a universal bilinear map for V and W, the target space V ⊗ W is the smallest such space in the precise sense given by the universal property: any other candidate target space receiving a universal bilinear map must be canonically isomorphic to V ⊗ W, so there is no smaller space that still satisfies universality, and no larger space is needed.

Visualizing the Target Space

Consequences of Treating It as a Target Space

Linear Maps Out of the Target Determine Bilinear Behavior

Because the target space is generated exactly by simple tensors under the relations of bilinearity, defining a linear map out of V ⊗ W is equivalent to defining a bilinear map out of V × W, and this equivalence is precisely the isomorphism Hom(V ⊗ W, Z) ≅ Bil(V × W, Z) central to the theory.

Change of Target Space Under Linear Substitution

If V and W are replaced by images under linear maps φ: V → V' and ψ: W → W', the target space transforms via the induced map φ ⊗ ψ: V ⊗ W → V' ⊗ W', showing that the target space construction is functorial and not merely a static receptacle for one fixed pair of factors.

Broader Significance

Foundation for Tensor Spaces in Applications

Understanding V ⊗ W as a target space clarifies why tensors used in physics and engineering, such as stress or inertia tensors, are represented as elements of a space whose dimension is the product of the dimensions of the spaces being combined, rather than their sum.

Basis for Iterated and Multilinear Target Spaces

The same target-space perspective extends to the iterated tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, which serves as the target of the universal multilinear map for n factors, with dimension equal to the product of the individual dimensions, generalizing the two-factor case described here.