1.2.35 Canonical Tensor Product Map Definition

The canonical tensor product map defines how tensors combine in algebra, forming a foundational structure for multilinear operations in tensor algebras.

Canonical Tensor Product Map Definition is the characterization of the specific bilinear, or more generally multilinear, map that pairs the factor spaces of a tensor product with the tensor product space itself, sending a tuple of vectors to their tensor product. This map, usually denoted $τ$ or written simply with the symbol $\otimes$ , is the piece of data that accompanies the tensor product space in its universal property, and it is called canonical because it is fixed by the construction of the tensor product itself rather than chosen arbitrarily.

Formal Definition

Let $V_{1}, \dots, V_{k}$ be vector spaces over a field $F$ , and let $V_{1} \otimes \dots \otimes V_{k}$ denote their tensor product space. The canonical tensor product map is the function

τ : V_{1} \times \dots \times V_{k} \to V_{1} \otimes \dots \otimes V_{k}

defined by

τ (v_{1}, \dots, v_{k}) = v_{1} \otimes \dots \otimes v_{k}

This map is multilinear, meaning it is linear in each argument $v_{i}$ separately when the remaining arguments are held fixed, a property that follows directly from the bilinearity relations imposed in the construction of the tensor product. The image of $τ$ is exactly the set of decomposable, or simple, tensors, and this image generates the entire tensor product space by finite linear combination, even though $τ$ itself is not linear and its image alone does not cover the whole space.

Role in the Universal Property

The canonical tensor product map is the fixed piece of structure against which the universal property is stated. Given any multilinear map

T : V_{1} \times \dots \times V_{k} \to U

the universal property guarantees a unique linear map $\tilde{T}$ on the tensor product space with

T = \tilde{T} \circ τ

so that $τ$ is precisely the map through which every multilinear map from the factor spaces must be routed. Without a fixed choice of $τ$ , the phrase "the tensor product satisfies the universal property" would be incomplete, since the property is a statement about the pair consisting of the space and this specific map, not about the space alone.

Naturality and Canonicity

The map $τ$ is called canonical because it arises directly from the construction of the tensor product and does not depend on auxiliary choices such as a basis of the factor spaces. This is distinct from many other useful maps associated with vector spaces — such as the isomorphism between a finite-dimensional space and its double dual, which is also canonical, versus the isomorphism between a space and its ordinary dual, which requires an arbitrary choice of basis or inner product. Because $τ$ is canonical in this sense, statements involving it, such as the universal property itself, hold uniformly across all tensor product constructions that are isomorphic via the unique isomorphism guaranteed by the universal property.

Properties Inherited from Multilinearity

Because $τ$ is multilinear rather than linear, it does not preserve addition in the ordinary sense; instead it distributes according to the bilinearity relations that define the tensor product,

τ (v_{1} + v_{1}^{'}, \dots, v_{k}) = τ (v_{1}, \dots, v_{k}) + τ (v_{1}^{'}, \dots, v_{k})

Consequently, the image of $τ$ is not itself a subspace, even though it spans one, and the failure of $τ$ to be additive across its full domain in the naive sense is exactly what distinguishes a multilinear map from a linear one, motivating the passage to the tensor product space in the first place.

Restriction to Elementary Tensors

Because $τ$ is defined directly on tuples of vectors, its output is by construction always an elementary, or decomposable, tensor: every value $τ (v_{1}, \dots, v_{k})$ is a single product term, never a sum of several such terms. Understanding $τ$ as the source of exactly the decomposable elements of the tensor product space clarifies why general, non-decomposable tensors require finite sums of values of $τ$ to be expressed, and it situates the canonical map as the bridge between the multilinear-algebra description of tensor products, given entirely in terms of vector tuples, and the linear-algebra description of the tensor product space as a vector space of its own.

Role Within Tensor Algebra

The canonical tensor product map is the concrete link between multilinear maps on the factor spaces and linear maps on the tensor product space throughout tensor algebra. Every statement about how multilinear forms, bilinear operations, or higher tensor constructions relate to the tensor product ultimately routes through this map: it is the object being universally factored through, the source of every elementary tensor, and the fixed reference point relative to which the canonical isomorphisms among differently constructed tensor product spaces are themselves defined.