4.17.1 Tensor Universal Property Input Map

Tensor Universal Property Input Map is the canonical multilinear map sending a tuple of vectors, one from each of several vector spaces, to their tensor product, serving as the fixed reference map through which every other multilinear map on those same spaces must factor according to the universal property of the tensor product. It is the specific map, often denoted with a small Greek letter such as ι or written simply with the tensor product symbol, that the universal property singles out as canonical, distinguishing it from the many other multilinear maps that could be defined on the same collection of vector spaces.

Defining the Input Map

The Map on a Product of Vector Spaces

For vector spaces V_1, ..., V_k, the universal property input map is the function

sending a tuple (v_1, ..., v_k) to the elementary tensor v_1 ⊗ v_2 ⊗ ⋯ ⊗ v_k in the tensor product space. This map is defined directly by the construction of the tensor product itself, and its output for any given tuple is, by definition, the symbol v_1 ⊗ ⋯ ⊗ v_k regarded as an element of the tensor product.

Multilinearity of the Input Map

The map ι is itself multilinear in the k arguments v_1, ..., v_k, satisfying both the tensor multilinear additivity property and the tensor multilinear homogeneity property in each slot separately, since these properties are built into the defining relations of the tensor product space: expressions such as (u + w) ⊗ v_2 ⊗ ⋯ and u ⊗ v_2 ⊗ ⋯ + w ⊗ v_2 ⊗ ⋯ are identified as equal elements of the tensor product by construction.

The Role of the Input Map in the Universal Property

Factoring Multilinear Maps Through the Input Map

The universal property of the tensor product states that for any vector space W and any multilinear map M : V_1 × ⋯ × V_k → W, there exists a unique linear map ℓ : V_1 ⊗ ⋯ ⊗ V_k → W such that

meaning that applying ι first, then ℓ, reproduces M exactly. The input map ι is the fixed piece of this factorization; only ℓ varies depending on the particular multilinear map M being represented.

Why the Input Map Cannot Be Replaced Arbitrarily

The universal property holds specifically for ι, not for an arbitrary multilinear map into the tensor product space; a different multilinear map ι' into the same tensor product space would not, in general, permit every multilinear map M to factor through it uniquely, since the defining relations of the tensor product are calibrated precisely to ι, the map sending tuples to their elementary tensor products.

The Input Map and Uniqueness of the Factorization

Uniqueness Follows from Surjectivity onto Elementary Tensors

The elementary tensors v_1 ⊗ ⋯ ⊗ v_k obtained as outputs of ι span the entire tensor product space, even though they do not exhaust it as a set; because they span, any linear map ℓ is completely determined by its values on outputs of ι, which is exactly the requirement that ℓ ∘ ι = M pins down ℓ uniquely, connecting the universal property directly to tensor multilinear extension construction applied to the spanning set of elementary tensors.

The Input Map as the Source of All Compatibility Conditions

Every compatibility condition that a linear map ℓ must satisfy in order to be well-defined on the tensor product space arises from relations among elementary tensors that are themselves consequences of the multilinearity of ι; the input map is thus the origin point from which every such relation, and hence every compatibility requirement in the general extension construction, ultimately derives.

The Input Map in Component Terms

Sending Basis Tuples to Basis Tensors

If each V_r has a basis, the input map sends a tuple of basis vectors, one from each V_r, to the corresponding basis tensor of the tensor product space, which is exactly how the standard basis of the tensor product space is constructed: every basis tensor arises as the image, under ι, of some tuple of basis vectors drawn from the individual spaces.

Non-Surjectivity onto the Full Tensor Product

Although the images of ι span the tensor product space, ι itself is not surjective onto the tensor product space in general, since a general element of the tensor product space is a sum of several elementary tensors and need not itself be expressible as a single elementary tensor v_1 ⊗ ⋯ ⊗ v_k; this distinction between spanning and surjectivity is central to understanding the structure of the tensor product beyond its simplest elements.

Diagrammatic Summary

The diagram shows the input map ι sending tuples into the tensor product space, through which any multilinear map M factors uniquely as ℓ ∘ ι, with ℓ the unique linear map guaranteed by the universal property.