1.3.5 Tensor Space Structure

Tensor Space Structure organizes multilinear relationships through vector spaces, enabling complex algebraic operations in physics and mathematics.

Tensor Space Structure is the recognition that the collection of all tensors of a fixed type (p, q) built from a vector space V is not merely a set of individual multilinear objects but itself forms a vector space, T^p_q(V), complete with its own operations of addition and scalar multiplication, its own basis, and its own well-defined dimension. Treating each T^p_q(V) as a vector space in its own right is what allows tensors to be added together, scaled by field elements, and manipulated using all of the standard tools of linear algebra applied one level up from the original space V.

Vector Space Axioms for T^p_q(V)

Addition of Tensors

Two tensors of the same type (p, q) can be added componentwise, relative to any fixed basis, and the result is again a tensor of type (p, q).

{(S + T)}_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = S_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} + T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}}

This addition is well defined independent of the chosen basis, since both S and T transform identically under a change of basis, and the sum inherits the same transformation law, confirming that it, too, is a genuine type (p, q) tensor.

Scalar Multiplication

A tensor T of type (p, q) can be scaled by any element c of the field F, multiplying every component by c, and the result remains a tensor of type (p, q).

{(c, T)}_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}} = c T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}}

Zero Tensor and Additive Inverses

The tensor whose every component equals zero in some, and hence every, basis serves as the additive identity of T^p_q(V), and the tensor obtained by negating every component of T serves as its additive inverse, satisfying all of the axioms required of a vector space over F.

Basis and Dimension of T^p_q(V)

Constructing a Basis

Given a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n of V*, the space T^p_q(V) has a natural basis consisting of every tensor product of p basis vectors and q dual basis covectors.

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

for every combination of index values i_1, ..., i_p, j_1, ..., j_q ranging from 1 to n, and every tensor in T^p_q(V) is a unique linear combination of these basis elements, with the coefficients given exactly by the tensor's components.

Dimension Formula

Since there are n^(p+q) distinct basis elements constructed this way, the dimension of T^p_q(V) as a vector space equals n^(p+q), matching the total component count established for tensors of that type.

dim (T_{q}^{p}, (V)) = n^{p + q}

Subspaces Within T^p_q(V)

Symmetric and Antisymmetric Subspaces

Within a fixed tensor space T^p_q(V), the tensors that are totally symmetric in their upper indices, or totally symmetric in their lower indices, form a linear subspace, since the sum of two symmetric tensors is symmetric and any scalar multiple of a symmetric tensor remains symmetric. The same holds for totally antisymmetric tensors, which form a separate subspace disjoint from the symmetric one except for the zero tensor.

Trace-Free Subspaces

For type (1, 1) tensors, the tensors whose full contraction, or trace, equals zero form a subspace of codimension one within T^1_1(V), since the trace operation is linear and its kernel is therefore a linear subspace.

Isomorphism with the Space of Multilinear Maps

Two Equivalent Realizations

The vector space T^p_q(V), built as an iterated tensor product, is naturally isomorphic as a vector space to the space of multilinear maps taking p covectors and q vectors to a scalar, with the isomorphism sending a decomposable tensor product v_1 ⊗ ... ⊗ ω_q to the multilinear map that evaluates each argument against the corresponding factor and multiplies the results. This isomorphism confirms that the vector space structure described here matches the multilinear map perspective used elsewhere in tensor theory, providing two equivalent but complementary views of the same underlying space.

Diagrammatic Summary

The diagram summarizes T^p_q(V) as an ordinary vector space of dimension n^(p+q), spanned by the basis tensor products formed from the chosen basis of V and its dual basis, with addition and scalar multiplication defined componentwise relative to that basis.