12.13.1 Tensor Projection Source Space

Tensor Projection Source Space is the foundational space where tensor projections originate, defining their structure and operational context.

Tensor Projection Source Space is the tensor space on which a projection operation is defined, namely the space from which a projection map π : T → T′ originates before it is applied, where T is typically a tensor product of several factor spaces and T′ is a lower-dimensional or structurally simpler space obtained by discarding, contracting, or collapsing some of that structure. The source space supplies the full ambient set of tensors — including all the components, index slots, and factor spaces — out of which the projection operation selects, retains, or reduces a designated portion.

Role of the Source Space in a Projection

Source Space as the Domain of Definition

A tensor projection operation is only meaningful once its source space is fixed, since the map π : T → T′ is defined pointwise on elements of T. Specifying π without specifying T leaves the operation ambiguous, because the same formula for π can act differently depending on which ambient tensor space its argument is assumed to range over.

Relation to the Target Space

π : T ​ ​ \to ​ ​ T'

The source space T and target space T′ are related by π being surjective onto T′ (or onto its image), so that every element of T′ arises as the projection of some element of the source space. The source space is always at least as structurally rich as the target space, since the projection operation's purpose is to reduce structure, not add it.

Typical Forms of the Source Space

Tensor Product of Several Factors

T = V_{1} \otimes V_{2} \otimes \dots \otimes V_{n}

In the projection onto a single factor, the source space is the full multi-factor tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, and the projection operation extracts, contracts, or evaluates against the remaining factors to produce an element of one distinguished factor Vᵢ.

Direct Sum Decompositions

When T decomposes as a direct sum T = T₁ ⊕ T₂, the source space for the projection onto T₁ is the entire sum T, not just T₁. The projection map sends t₁ + t₂ ↦ t₁, and this formula depends on T being presented with its direct-sum decomposition — the source space carries that decomposition as part of its data, not merely its underlying set.

Higher-Rank Tensor Spaces Before Index Contraction

When projection is realized as contraction against a fixed covector or fixed index slot, the source space is the full rank-k tensor space V^{⊗k} (or its appropriate mixed variance version), and the projection reduces it to a rank-(k−1) tensor space by summing out one index against the fixed contracting element.

Diagram of the Source Space Feeding the Projection

Structural Requirements on the Source Space

Well-Defined Vector or Module Structure

The source space must itself be a well-defined vector space or module, complete with its own addition and scalar multiplication, so that the projection operation π can be required to be linear. A projection is characterized in part by respecting this structure: π(t + t′) = π(t) + π(t′) and π(c·t) = c·π(t) for scalars c, and both sides of these equations are only meaningful once the source space's own operations are fixed.

Idempotency Relative to the Source Space

When the projection is an endomorphism of a single space, π : T → T, the source and target spaces coincide, and the projection is additionally required to satisfy π ∘ π = π. This idempotency condition is stated entirely relative to the source space T: it asserts that applying π a second time to any element already produced by π, an element of the source space, changes nothing further.

Compatibility with Any Chosen Basis

If the source space has a chosen basis adapted to the projection — for instance, a basis split according to a direct-sum decomposition, or a product basis built from bases of the tensor factors — the projection is typically defined coordinate-wise relative to that basis. The source space's basis is therefore part of the data needed to write down the projection explicitly, even though the projection itself, as a linear map, exists independently of any particular basis choice.

Consequences of an Ambiguous Source Space

Ill-Defined Contraction Index

If the source space's tensor rank or factor decomposition is not fixed, an instruction to "project out the third factor" or "contract the second index" becomes ambiguous, since different presentations of what appears to be the same tensor may disagree on which slot is the third factor. Fixing the source space resolves this by fixing the ordering and identity of every factor or index slot in advance.

Mismatched Composition of Projections

Composing two projection operations, π₂ ∘ π₁, requires the target space of π₁ to coincide with the source space of π₂. An unclear or shifting notion of source space makes it impossible to verify this compatibility, which can produce a composite operation that is not actually well-defined as a single projection.