2.15.4 Tensor Finite Tensor Space Construction

Tensor Finite Tensor Space Construction builds finite-dimensional tensor spaces by systematically organizing tensor products within a structured algebraic framework.

Tensor Finite Tensor Space Construction is the procedure by which the space of type (p, q) tensors over a finite-dimensional vector space is built up as an algebraic object in its own right, starting from copies of the underlying space and its dual and combining them through the tensor product operation until a single, well-defined vector space emerges. This construction is what turns "a tensor" from an informal idea about multi-indexed arrays into a rigorously defined mathematical object with its own basis, dimension, and vector-space structure, and it is the finite-dimensional setting in which the construction behaves most simply, since every step terminates in a finite basis and a finite dimension.

Starting Materials

The Underlying Space and Its Dual

The construction begins with a finite-dimensional vector space V over a field F, together with its dual space V*, the space of linear functionals V → F. In the finite-dimensional setting, V* has the same dimension as V, and choosing a basis for V automatically produces a corresponding dual basis for V* through the pairing condition that each dual basis covector returns 1 on its matching basis vector and 0 on the others.

The Tensor Product of Two Spaces

The elementary building block is the tensor product of two vector spaces, V ⊗ W, defined as the vector space spanned by formal symbols v ⊗ w subject to bilinearity relations: (v_1 + v_2) ⊗ w = v_1 ⊗ w + v_2 ⊗ w, v ⊗ (w_1 + w_2) = v ⊗ w_1 + v ⊗ w_2, and scalar multiples pass freely across the ⊗ symbol. This space is characterized by a universal property: every bilinear map out of V × W factors uniquely through V ⊗ W.

Iterating the Construction

Building the Type (p, q) Space

The full tensor space T^p_q(V) is constructed by iterating the two-factor tensor product p times over V and q times over V*, then taking the tensor product of all the resulting factors together.

T_{q}^{p} (V) = \otimes_{i = 1}^{p} V \otimes \otimes_{j = 1}^{q} V^{*}

Because the tensor product operation is associative and commutative up to canonical isomorphism, the order in which the individual factors are combined does not affect the resulting space, only how its elements are labeled.

Associativity and Canonical Isomorphism

The statement that (V ⊗ W) ⊗ U is canonically isomorphic to V ⊗ (W ⊗ U) is what licenses writing the iterated tensor product without parentheses. This isomorphism sends (v ⊗ w) ⊗ u to v ⊗ (w ⊗ u) and is the reason the construction can be described as producing a single space T^p_q(V) rather than a tree of nested products.

Producing a Basis From the Construction

Basis of the Tensor Product

Given a basis e_1, ..., e_n of V and the dual basis e^1, ..., e^n of V*, the construction yields a basis for T^p_q(V) consisting of every possible tensor product of p basis vectors and q dual basis vectors, chosen with repetition and in order.

e_{i_{1}} \otimes \dots \otimes e_{i_{p}} \otimes e^{j_{1}} \otimes \dots \otimes e^{j_{q}}

Each choice of index tuples (i_1, ..., i_p) and (j_1, ..., j_q), with every index ranging from 1 to n, gives one basis element, and every tensor in T^p_q(V) is a unique linear combination of these basis elements.

Well-Definedness of the Construction

Because the construction depends only on V and its dimension, and not on any particular basis chosen along the way, the resulting space T^p_q(V) is intrinsic to V: switching to a different basis of V produces a different labeling of the same abstract space, related by the standard change-of-basis transformation law.

Diagrammatic View of the Assembly

The diagram shows the raw materials on top, p copies of V and q copies of V*, feeding into the single assembled space T^p_q(V) below, matching the formal definition of the iterated tensor product.

Finite Dimension as a Simplifying Feature

Why Finiteness Matters Here

Because V and V* are each finite-dimensional, every step of the construction produces another finite-dimensional space, the tensor product of finite-dimensional spaces being finite-dimensional with dimension equal to the product of the factors' dimensions. This guarantees that T^p_q(V) has a finite basis, that every tensor is a finite linear combination of basis tensors, and that no completion or limiting procedure is needed to make sense of the space, in contrast to constructions built over infinite-dimensional spaces.