12.19 Tensor Operation Closure Property

Tensor Operation Closure Property is the general question, asked separately of each tensor algebra operation, of whether applying that operation to inputs drawn from a specified space or type of tensors always produces an output that remains within that same specified space or type, as opposed to producing a tensor of a different rank, a different variance, or belonging to an altogether different tensor space. It functions as the organizing question behind the individual closure facts established for particular operations — tensor addition closure and scalar multiplication closure being the two most elementary instances — and it is what determines whether a given collection of tensors, equipped with a given operation, forms a self-contained algebraic structure or merely a starting point from which the operation departs into a larger ambient collection.

The General Closure Question

Formal Statement

Given a tensor operation op mapping a designated input collection X of tensors to some collection Y of results, closure asks whether Y coincides with X itself — whether the operation, applied to elements already inside X, never produces anything lying outside X. This is a question that must be posed and answered separately for every operation and every choice of X, since the same operation can be closed relative to one collection of tensors and not closed relative to another.

Distinguishing Closure from Mere Well-Definedness

An operation can be perfectly well-defined — producing a valid tensor as output for every valid input — while still failing to be closed on a particular collection X, simply because its output happens to belong to a different collection Y ≠ X. Closure is therefore a stronger, more specific claim than well-definedness: it additionally requires the output to land back inside the same collection the input came from.

Operations That Are Closed

Addition and Scalar Multiplication on a Fixed Type

The tensor addition closure property and the tensor scalar multiplication closure property jointly establish that, for any fixed variance type (p,q) and fixed space V, both of these operations remain entirely within the space of type-(p,q) tensors on V. These are the paradigm examples of tensor operation closure, and their joint truth is exactly what allows the collection of type-(p,q) tensors on V to be treated as a vector space.

Direct Sum Formation Within a Fixed Ambient Ring

If two subspaces A and B of a common ambient space C are combined via direct sum and their sum is itself contained in C, the injection-projection pair associated with that decomposition remains closed relative to C, since neither the inclusion nor the projection ever produces an object outside C's own tensor structure.

Operations That Are Not Closed

The Tensor Product Increases Type

The tensor product of two type-(p,q) tensors produces a tensor of type (2p,2q), not type (p,q), so the tensor product operation is emphatically not closed on the space of type-(p,q) tensors, even though it is a perfectly well-defined operation overall. Closure fails here not because the operation is invalid, but because its output systematically lands in a different, larger tensor space than its inputs.

Contraction Decreases Rank

Contracting a matched pair of slots on a type-(p,q) tensor produces a tensor of type (p−1,q−1), again departing from the original type. Contraction is therefore not closed on a fixed type-(p,q) space, mirroring the tensor product's failure of closure but in the opposite direction of rank change.

Pullback and Pushforward Move Between Different Spaces

The pullback and pushforward operations are, by their very construction, designed to move tensors from one space to a different space via a source map f; they are never closed on a single space V unless f happens to be an automorphism of V onto itself, in which case the operation coincidentally returns to the same space it started from.

Diagram of Closed versus Non-Closed Operations

Consequences of Closure or Its Absence

Closure Enables Algebraic Structure

When an operation is closed on a given collection, that collection together with the operation forms a self-contained algebraic structure — a vector space under addition and scalar multiplication, or a monoid or group under a suitably closed composition — permitting the full machinery of that algebraic structure's theory to be applied directly.

Non-Closure Motivates Larger Ambient Structures

When an operation is not closed on a given collection, the natural response is to enlarge the collection to one where closure does hold — for instance, forming the full tensor algebra T(V) = ⊕ₖ V^⊗ᵏ specifically so that the tensor product, which is not closed on any single fixed rank, becomes closed on this larger graded direct sum, since multiplying tensors of any two ranks always lands back inside the union of all ranks.

Verifying Closure for a New Operation

Checking the Output Type Against the Input Type

The standard method for verifying closure is to determine the variance type and ambient space of an operation's output purely from the variance types and ambient spaces of its inputs, using the operation's own defining formula, and then comparing this output type and space against the collection in question.

Closure as a Prerequisite for Further Structural Claims

Before asserting that a certain collection of tensors equipped with a certain operation satisfies some richer algebraic property — forming a ring, an algebra, or a module — closure of that operation on that collection must be established first, since none of these richer structural claims are meaningful if the operation is capable of producing results falling outside the collection under discussion.