4.16.5 Tensor Extension Tensor Product Preparation

Tensor Extension Tensor Product Preparation is the preliminary construction of the tensor product space itself, built as a quotient of a free vector space on formal tuples by exactly the relations that multilinearity requires, carried out before any particular tensor multilinear extension construction can be applied. It supplies the ambient space and the canonical tensor universal property input map that every subsequent extension argument relies upon, making explicit the construction that is typically taken for granted whenever a multilinear map is said to correspond to a linear map on a tensor product.

Building the Free Vector Space on Tuples

Formal Linear Combinations of Tuples

Given vector spaces V_1, ..., V_k, the preparation begins by forming the free vector space F(V_1 × ⋯ × V_k) on the set of all tuples (v_1, ..., v_k), treating each tuple as an independent formal basis symbol, with no relation yet imposed between, for example, (u + w, v_2, ..., v_k) and the separate symbols (u, v_2, ..., v_k) and (w, v_2, ..., v_k).

Why the Free Space Is Too Large

This free vector space, by itself, is far larger than any tensor product space needs to be, since it treats every tuple as linearly independent of every other, including tuples that ought to be related by the additivity and homogeneity expected of a multilinear structure; the free space is only the raw material from which the tensor product is prepared, not the tensor product itself.

Imposing the Multilinearity Relations

The Subspace of Relations to Quotient By

The preparation continues by forming the subspace R of the free vector space, spanned by all elements of the form

together with all elements of the form (⋯, λv, ⋯) - λ(⋯, v, ⋯), for every slot, every pair of vectors u, w, every scalar λ, and every fixed choice of the remaining entries. These are exactly the relations that tensor multilinear additivity and tensor multilinear homogeneity require of any multilinear map.

Defining the Tensor Product as a Quotient

The tensor product space is then defined as the quotient

and the tensor universal property input map ι sends a tuple to the coset of its corresponding free basis symbol modulo R, which is exactly the elementary tensor v_1 ⊗ ⋯ ⊗ v_k.

Why This Preparation Makes Multilinear Extension Possible

Automatic Multilinearity of the Input Map

Because tuples related by the additivity and homogeneity relations become equal, by construction, once passed to the quotient, the input map ι is automatically multilinear; this is precisely the property needed before the tensor universal multilinear property can even be stated, since the universal property presupposes that ι itself is a valid multilinear map.

Elementary Tensors as a Spanning Set for Extension

Since the free vector space is spanned by all tuples, and the tensor product is a quotient of the free space, the images of all tuples under ι, the elementary tensors, span the tensor product space; this spanning set is exactly what tensor multilinear extension construction requires as its starting point, and it is delivered directly by the preparation described here rather than assumed separately.

Preparation as the Origin of Extension Compatibility

Compatibility Guaranteed by Construction, Not Assumed

Because the relations defining R are exactly the relations tensor extension compatibility requires to hold, any assignment of values to tuples that respects multilinearity automatically descends to a well-defined map on the quotient; the preparation of the tensor product is what guarantees, once and for all, that compatibility never fails for data arising from an actual multilinear map, removing the need to verify compatibility case by case for every application of the extension construction.

Preparation as a One-Time Cost

This construction of the tensor product space, and the accompanying verification that its defining relations match the multilinearity conditions exactly, needs to be carried out only once for a given collection of vector spaces; every subsequent invocation of tensor extension construction on those same spaces can rely on the tensor product having already been prepared correctly.

Relation to Later Extension Arguments

Supplying the Ambient Space for Extension

Every extension argument used elsewhere, including tensor multilinear extension from basis and the more general tensor extension uniqueness, takes place within the tensor product space assembled by this preparation; without this prior construction, there would be no fixed ambient space in which the extended tensor could be said to live.

Preparation Precedes but Does Not Replace Extension

Preparing the tensor product space establishes the space and the canonical input map, but it does not by itself determine any particular tensor; a specific tensor still requires prescribed values, checked for compatibility if necessary, and only then extended by the methods of tensor multilinear extension construction within the space this preparation has made available.

Diagrammatic Summary

The diagram shows the free vector space on tuples being quotiented by the subspace of multilinearity relations, producing the prepared tensor product space in which every subsequent extension construction takes place.