2.3.4 Tensor Vector Space Closure Structure

Tensor Vector Space Closure Structure extends vector spaces with tensor algebras, enabling multilinear operations and broader algebraic structures.

Tensor Vector Space Closure Structure is the property of the set of tensors of a fixed type that guarantees the result of applying tensor addition or scalar action to elements of that set always remains within the same set. It is the structural fact that underlies the entire vector space character of a tensor space: without closure, addition and scalar action would risk producing objects outside the space under study, and no consistent linear algebra could be built on top of them.

Formal Statement

Setting

Let $V$ be a finite-dimensional vector space over a field $F$ , and let $T_{s}^{r} (V)$ be the set of all tensors of type $(r, s)$ over $V$ , that is, multilinear maps

T : \underset{r factors}{V^{*} \times \dots \times V^{*}} \times \underset{s factors}{V \times \dots \times V} \to F

Closure under Addition

For any two tensors $S, T \in T_{s}^{r} (V)$ , the sum $S + T$ is again a member of $T_{s}^{r} (V)$ :

S, T \in T_{s}^{r} (V) \Rightarrow S + T \in T_{s}^{r} (V)

Closure under Scalar Action

For any scalar $α \in F$ and any tensor $T \in T_{s}^{r} (V)$ , the scaled tensor $α T$ remains in $T_{s}^{r} (V)$ :

α \in F, T \in T_{s}^{r} (V) \Rightarrow α T \in T_{s}^{r} (V)

Why Closure Holds

Multilinearity is Preserved

Closure follows directly from the definitions of the two operations. The pointwise sum of two multilinear maps is itself multilinear in each argument, and a scalar multiple of a multilinear map is likewise multilinear, since fixing all but one argument still leaves a linear function of the remaining one after adding or scaling. No new type of object is produced by either operation; only maps of the same domain and codomain, satisfying the same multilinearity conditions, result.

Fixed Rank and Valence

Both operations act without altering the number of vector arguments or covector arguments a tensor accepts. Addition and scalar action modify only the output values of the multilinear map, never its input signature, so the rank $r$ and the covariant order $s$ are preserved exactly, keeping the result within the same type $(r, s)$ .

Componentwise Verification

Component Behavior

In a fixed basis, a tensor is represented by components $T_{j_{1} \dots j_{s}}^{i_{1} \dots i_{r}}$ , indexed over the same ranges regardless of which tensor of type $(r, s)$ is being described. Adding or scaling components produces an array indexed over exactly the same ranges, so the result is automatically a valid component array for a tensor of the identical type, verifying closure at the level of coordinates.

Consistency Across Bases

Because the component transformation law for type $(r, s)$ tensors is linear in the components, applying addition or scalar action commutes with a change of basis. The closure property therefore holds simultaneously in every basis, and is not an artifact of a particular coordinate choice.

Consequence: Subspace Structure

Embedding in a Larger Space

If tensors of varying types are considered together inside a larger direct sum $\underset{r, s}{\oplus} T_{s}^{r} (V)$ , closure of each fixed-type piece under addition and scalar action shows that each $T_{s}^{r} (V)$ forms a linear subspace of that larger space, rather than merely a subset.

Nonexistence of Type Mixing

Closure structure also clarifies what tensor addition and scalar action cannot do: since the result of either operation never leaves the type $(r, s)$ , these operations provide no mechanism for combining tensors of different types into a single object of a third type. Such combinations require other constructions, such as the tensor product, rather than addition or scalar action.

Role in the Vector Space Axioms

Prerequisite for Group and Field Axioms

Closure is listed first among the vector space axioms because every other axiom, commutativity, associativity, existence of a zero tensor, existence of additive inverses, and compatibility with scalar multiplication, presupposes that the operations in question actually produce elements of the same set on which those axioms are asserted. Without closure, statements like $S + T = T + S$ would not even be well-posed, since the two sides might otherwise fail to lie in a common space.

Finite Dimensionality

Given a basis of $V$ of dimension $n$ , the tensor space $T_{s}^{r} (V)$ has dimension $n^{r + s}$ , and closure under addition and scalar action confirms that this finite-dimensional coordinate description is complete and self-contained: every linear combination of tensors of type $(r, s)$ , with coefficients in $F$ , again yields a tensor expressible in the same $n^{r + s}$ -dimensional coordinate system.