5.17.2 Tensor Product Real Field Context

Explore how tensor products operate within the real field, establishing foundational context for algebraic structures and their applications.

Tensor Product Real Field Context is the specific instance of the tensor product construction in which the scalar field F is taken to be the field of real numbers ℝ, making V and W real vector spaces and V ⊗ W a real vector space of dimension dim(V) · dim(W). This context is the setting most commonly encountered in applied mathematics, physics, and engineering, where tensors are used to model geometric and physical quantities defined over ordinary real-valued coordinates, and it carries specific features, such as compatibility with real inner products and orderings, that are not automatically present when working over an arbitrary field.

The Real Field as Scalars

Basic Setup

In this context, V and W are vector spaces over ℝ, meaning scalar multiplication is defined by real numbers, and the tensor product V ⊗ W is constructed exactly as in the general case, with the free module F(V × W) taken over ℝ and the relation submodule R generated by the standard bilinear identities using real scalar coefficients.

Ordinary Real-Number Homogeneity

The homogeneity relations in this context take the familiar form, with c ranging over all real numbers:

(c v) \otimes w = c (v \otimes w) = v \otimes (c w), c \in ℝ

Additional Structure Available in the Real Context

Compatibility with Inner Products

If V and W are real inner product spaces with inner products ⟨·,·⟩_V and ⟨·,·⟩_W, the tensor product V ⊗ W inherits a natural inner product defined on simple tensors by:

⟨ v \otimes w, v^{'} \otimes w^{'} ⟩ = ⟨ v, v^{'} ⟩ \cdot_{V} ⟨ w, w^{'} ⟩ \cdot_{W}

extended bilinearly to all of V ⊗ W. This inner product does not have a canonical analogue when the scalar field lacks a compatible notion of positivity, making it a distinctive feature of the real (and more generally ordered or metrized) context.

Orthonormal Basis Tensors

If {eᵢ} and {fⱼ} are orthonormal bases of V and W with respect to their real inner products, the induced basis {eᵢ ⊗ fⱼ} is automatically orthonormal in V ⊗ W under the inherited inner product, a convenient property exploited throughout numerical linear algebra involving tensor spaces.

Real Field Context in Applications

Physical Tensors as Real Tensor Products

Physical quantities such as the stress tensor, the moment of inertia tensor, and the metric tensor of a real Riemannian manifold are all elements of tensor products of real vector spaces, typically the tangent space of a manifold and its dual, and rely on the real field context both for their construction and for the geometric interpretation of quantities like magnitude and orthogonality.

Real Matrices as Tensor Products of Real Vector Spaces

Under a choice of bases, the tensor product V ⊗ W of two finite-dimensional real vector spaces is identified with the space of real matrices Mat(dim V, dim W; ℝ), connecting the abstract real tensor product context directly to ordinary linear algebra over the reals.

Diagram of the Real Context

Distinctions from Other Field Contexts

Absence of Complex Conjugation Considerations

Unlike the complex field context, where a choice must be made between the bilinear tensor product and the conjugate-linear (sesquilinear) tensor product used for Hermitian inner products, the real field context has no such ambiguity, since real scalar multiplication has no distinct conjugate operation.

Real Tensor Products versus Complexification

A real tensor product V ⊗_ℝ W is generally distinct from the tensor product V_ℂ ⊗_ℂ W_ℂ of the complexified spaces, and passing between the two requires an explicit extension-of-scalars construction, underscoring that the real field context is not simply a restriction of the complex case but a self-contained setting with its own construction.

Significance of Fixing the Real Context

Default Assumption in Most Applied Treatments

Because the vast majority of applied uses of tensors, in physics, engineering, and classical differential geometry, take place over the reals, many textbooks state tensor product theory assuming the real field context implicitly, without repeating the general field-independent construction each time.

Foundation for Further Specialization

The real field context serves as the base setting from which further specialized structures are built, such as tensor products equipped with a real inner product, real tensor algebras used in continuum mechanics, and real symmetric or antisymmetric tensor powers used in the study of differential forms.