12.15.3 Tensor Direct Sum Space Construction

Tensor Direct Sum Space Construction combines vector spaces into a larger space, preserving individual structures while enabling operations across components.

Tensor Direct Sum Space Construction is the explicit procedure by which the direct sum V₁ ⊕ V₂ ⊕ ... ⊕ Vₙ is built as a genuine vector space from a given list of tensor spaces, together with the verification that the resulting set of tuples, equipped with componentwise operations, satisfies every vector space axiom. It is the foundational construction underlying the tensor direct sum operation, establishing that "stacking spaces together" produces a mathematically legitimate object before any of that object's further properties — dimension additivity, canonical inclusions, canonical projections — can be meaningfully discussed.

Building the Underlying Set

Cartesian Product as Raw Material

V_{1} \times V_{2} \times \dots \times V_{n} = {(v_{1}, \dots, v_{n}) : v_{k} \in V_{k}}

The construction begins with the ordinary set-theoretic Cartesian product of the underlying sets of V₁, ..., Vₙ. At this stage, the product is merely a set of tuples with no algebraic structure; the direct sum space construction consists of equipping this set with operations and then verifying those operations behave correctly.

Defining Componentwise Addition

(v_{1}, \dots, v_{n}) + ({v^{'}}_{1}, \dots, {v^{'}}_{n}) = (v_{1} + {v^{'}}_{1}, \dots, v_{n} + {v^{'}}_{n})

Addition of two tuples is defined by adding corresponding entries using each summand's own addition operation. Since each Vₖ already has a well-defined addition, this rule is well-defined for every pair of tuples, and produces another tuple in the same product set.

Defining Scalar Multiplication

Scalar multiplication is likewise defined componentwise, c·(v₁, ..., vₙ) = (c·v₁, ..., c·vₙ), applying each summand's own scalar action independently in every slot. As with addition, this rule is immediately well-defined because it only ever invokes operations already known to exist within each Vₖ.

Verifying the Vector Space Axioms

Closure

Because addition and scalar multiplication are defined entry by entry using each Vₖ's own closed operations, the result of adding two tuples or scaling one tuple is again a tuple with entries in the correct respective spaces, confirming closure of both operations on the product set.

Associativity, Commutativity, and Distributivity

Each of these axioms reduces, entry by entry, to the corresponding axiom already holding in the individual space Vₖ. For instance, associativity of addition on the direct sum, (u + v) + w = u + (v + w), holds because (uₖ + vₖ) + wₖ = uₖ + (vₖ + wₖ) holds in each Vₖ separately; verifying the direct sum's axioms is therefore never more than restating the summands' own axioms slot by slot.

Existence of Zero and Additive Inverses

0 = (0_{1}, \dots, 0_{n})

The zero element of the direct sum is the tuple of zero elements from each summand, and the additive inverse of (v₁, ..., vₙ) is (−v₁, ..., −vₙ), again built entirely from data already guaranteed to exist within each Vₖ. No new construction is required beyond assembling these known elements into a single tuple.

Diagram of the Construction Process

Construction with Additional Compatible Structure

Extending to Direct Sums of Algebras

If each Vₖ is additionally an algebra with its own multiplication ·ₖ, the direct sum space construction extends by defining (v₁, ..., vₙ)·(v₁′, ..., vₙ′) = (v₁·₁v₁′, ..., vₙ·ₙvₙ′). Verifying this makes the direct sum an algebra follows the identical entry-by-entry pattern used for the vector space axioms, checking associativity and distributivity of multiplication one summand at a time.

Extending to Direct Sums Equipped with Bilinear Forms

If each Vₖ carries a bilinear form βₖ, one canonical extension defines a form on the direct sum by β((v₁,...,vₙ),(v₁′,...,vₙ′)) = Σₖ βₖ(vₖ, vₖ′), summing the individual forms across summands. This produces a bilinear form on the whole direct sum built entirely from the given data, with cross terms between different summands set to zero by convention.

Uniqueness of the Resulting Construction

Independent of Any Auxiliary Choices

The direct sum space construction requires no arbitrary choices — no basis, no ordering beyond the fixed index labeling of the summands, and no auxiliary embedding — since every operation is defined directly and canonically from the summands' own existing structure. Two mathematicians constructing V₁ ⊕ V₂ independently, given the same V₁ and V₂, necessarily arrive at the identical space.

Universal Property Characterization

The constructed direct sum can equivalently be characterized, up to canonical isomorphism, by a universal property: it is the unique space V equipped with inclusions ιₖ : Vₖ → V such that any collection of linear maps fₖ : Vₖ → M into a common space M factors uniquely through a single linear map V → M agreeing with each fₖ on the corresponding summand. This universal property confirms that the explicit tuple-based construction described above is not an arbitrary choice of model but the essentially unique object satisfying the direct sum's defining role among tensor spaces.