12.15 Tensor Direct Sum Operation

Tensor Direct Sum Operation is the operation that combines two or more tensor spaces V₁, V₂, ..., Vₙ into a single larger space V₁ ⊕ V₂ ⊕ ... ⊕ Vₙ whose elements are ordered tuples (v₁, v₂, ..., vₙ) with componentwise addition and scalar multiplication, and which simultaneously furnishes a canonical family of inclusion maps embedding each summand into the whole and a canonical family of projection maps recovering each summand back out of the whole. It is the structural framework that makes precise the informal idea of "stacking" several tensor spaces side by side without mixing their contents, in contrast to the tensor product operation, which instead multiplies spaces together and generally increases dimension multiplicatively rather than additively.

Construction of the Direct Sum

Underlying Set and Operations

Addition and scalar multiplication on V₁ ⊕ V₂ are defined componentwise: (v₁, v₂) + (v₁′, v₂′) = (v₁ + v₁′, v₂ + v₂′) and c·(v₁, v₂) = (c·v₁, c·v₂). Every operation on the direct sum reduces to the corresponding operation performed independently within each summand, with no interaction between the components.

Dimension Additivity

For finite-dimensional summands, the dimension of the direct sum is the sum of the individual dimensions, obtained by concatenating a basis of V₁ with a basis of V₂ (placed in the two respective slots). This additive dimension behavior is the defining numerical signature distinguishing the direct sum from the tensor product, whose dimension is instead multiplicative.

Canonical Inclusions and Projections

The Inclusion of Each Summand

The direct sum construction produces, for each summand Vₖ, a canonical tensor inclusion operation ιₖ : Vₖ → V₁ ⊕ ... ⊕ Vₙ, placing Vₖ into its designated slot and setting every other slot to zero. Each ιₖ is automatically linear and injective, satisfying the embedding map role without any further verification needed beyond the construction itself.

The Projection onto Each Summand

Dually, the direct sum produces canonical tensor projection operations πₖ : V₁ ⊕ ... ⊕ Vₙ → Vₖ, extracting the k-th slot and discarding the rest. Each πₖ is linear and idempotent when viewed as an endomorphism composed with ιₖ, satisfying the requirements of a tensor projection operation directly by construction.

Retraction and Orthogonality Relations

The inclusions and projections satisfy the retraction identity for matching indices and vanish for mismatched indices, and the sum Σₖ ιₖ ∘ πₖ equals the identity map on the whole direct sum. These identities package together the structure-preservation and idempotency properties studied separately for general inclusions and projections into one coherent family.

Diagram of a Direct Sum with Its Inclusions and Projections

Extension of the Direct Sum to Multilinear Structures

Direct Sums of Tensor Algebras

If A₁ and A₂ are algebras (each equipped with its own multiplication), the direct sum A₁ ⊕ A₂ extends to an algebra by defining (a₁, a₂)·(a₁′, a₂′) = (a₁a₁′, a₂a₂′), multiplying componentwise and never mixing terms from different summands. This gives a canonical way to build a larger algebra out of two smaller ones whenever multiplicative interaction between them is not desired.

Distributivity over Tensor Products

The tensor product distributes over the direct sum up to canonical isomorphism, meaning tensoring a direct sum with another space produces a direct sum of the individual tensor products. This distributivity is what allows a tensor space built from several direct-sum pieces to be analyzed factor by factor after tensoring, rather than requiring the whole sum to be tensored as an undivided unit.

Distinguishing Direct Sum from Direct Product

Finite Case Coincidence

When only finitely many summands are involved, the direct sum V₁ ⊕ ... ⊕ Vₙ coincides exactly with the direct product V₁ × ... × Vₙ, since both consist of all tuples (v₁, ..., vₙ) with no restriction. The distinction between the two constructions is invisible in this finite setting.

Infinite Case Divergence

For infinitely many summands {Vᵢ}_{i∈I} with I infinite, the direct sum consists only of tuples with all but finitely many entries equal to zero, while the direct product allows arbitrary tuples with no such restriction. The direct sum is therefore, in general, a proper subspace of the direct product once infinitely many summands are combined, and the tensor direct sum operation specifically refers to this finitely-supported construction.