2.12.1 Tensor Vector Addition Closure

Tensor Vector Addition Closure is the guarantee that adding any two vectors from a given vector space always produces another vector belonging to that same space, so the addition operation never escapes the boundaries of the space it is defined on. Closure under addition is one of the two operation-based conditions, alongside closure under scalar multiplication, that any subset must satisfy to qualify as a subspace, and it is a baseline requirement for the vector space itself.

Formal Statement

Closure as Part of the Vector Space Definition

For any two vectors drawn from the vector space, their sum is guaranteed to be another vector within that same vector space.

Closure Restricted to a Subspace

When considering a subset intended to be a subspace, closure under addition means that summing two elements of the subset always yields an element still inside the subset, not merely inside the larger ambient space.

Role in Testing Subspaces

One of the Standard Subspace Criteria

Closure under addition, together with zero vector membership and closure under scalar multiplication, forms the standard three-part test used to determine whether a given subset of a vector space is itself a subspace.

Failure of Closure Disqualifies a Subset

If even a single pair of elements from a subset produces a sum lying outside that subset, the subset fails the closure test and cannot be a subspace, regardless of how many other subspace-like properties it might otherwise appear to satisfy.

Closure Behavior in Coordinates

Entrywise Closure of Coordinate Vectors

Since vector addition corresponds to entrywise addition of coordinate tuples, closure under addition guarantees that summing the coordinate vectors of two elements from the space produces a coordinate tuple that itself corresponds to a valid vector in the space.

Consistency Across Any Basis

Because closure is a property of the abstract vector space, it holds regardless of which basis is used to express vectors in coordinates, so the closure guarantee is not tied to any particular coordinate system.

Role in Tensor Construction

Ensuring Tensor Sums Remain Valid Tensors

Closure under addition within each factor vector space is what ultimately supports the fact that adding two tensors built from those factors produces another valid tensor within the same tensor space, preserving the algebraic structure at the level of tensors.

Interaction With Span Closure

Vector addition closure is directly linked to span closure, since the closure of a span under addition, established separately for spans, is itself an instance of this more general addition closure property applied to the specific subset formed by the span.

Summary of Key Properties

No Escape From the Vector Space Under Addition

Tensor Vector Addition Closure guarantees that the result of adding any two vectors always remains within the same vector space or subspace under consideration.

Foundational Test for Structural Validity

This closure property functions as a foundational and frequently applied test throughout tensor algebra, used whenever a new subset or construction needs to be verified as respecting the vector space's additive structure.