11.14.3 Tensor Contravariant Slot Vector Space Relation

Tensor Contravariant Slot Vector Space Relation explains how contravariant slots link to vector spaces via coordinate transformations and algebraic structure.

Tensor Contravariant Slot Vector Space Relation is the identification of a tensor's contravariant slot with an element of the double dual space, which is naturally identified with the original vector space itself, meaning that each upper index of a tensor, considered alone, behaves precisely like a vector in its capacity to be evaluated on covectors, which is the defining property recovered through the natural double dual identification.

Definition and Setting

The Double Dual and Its Natural Identification With the Original Space

For finite-dimensional vector spaces, the double dual space, meaning the dual of the dual space, is naturally identified with the original vector space, and a single contravariant slot of a tensor, viewed in isolation with all other slots held fixed, is exactly an element of this double dual, which is to say an ordinary vector acting on covectors.

v (ω) = ω (v) = v^{i} ω_{i}

Contravariant Components as Vector Space Coordinates

The contravariant components of a tensor, when restricted to a single slot, can be read as the coordinates of an element of the original vector space relative to the ordinary basis, since specifying these components completely determines the vector that the slot represents, once acting on covectors through evaluation.

Structural Consequences of the Vector Space Identification

Explaining the Origin of the Contravariant Transformation Law

Identifying a contravariant slot with an element of the vector space explains why its components transform with the direct Jacobian factor, since this is precisely the transformation law obeyed by ordinary basis vectors under change of coordinates, and the coordinates of a fixed vector must transform in the compensating manner relative to that basis.

T^{i^{'}} = \frac{\partial x^{i^{'}}}{\partial x^{j}} T^{j}

Consistency With the Pairing of Covectors and Vectors

Because a contravariant slot represents an element of the vector space, evaluating it on a covector reproduces exactly the pairing operation between a vector and a covector, reinforcing that filling a contravariant slot with a covector is the correct and only meaningful way to obtain a scalar from that slot.

Extension to Tensors With Multiple Contravariant Slots

Multiple Vector Space Elements Combined

A tensor with several contravariant slots corresponds to a multilinear map built from several elements of the vector space acting together, one per slot, so that the tensor as a whole can be understood as an element of a tensor product of copies of the original vector space.

Preserving the Vector Space Interpretation Under Partial Evaluation

If some but not all contravariant slots of a multi-slot tensor are filled with covectors, the remaining unfilled slots still retain their vector space character, so that the partially evaluated object continues to behave as an element of the vector space in each of its remaining upper index positions.

Role Within Tensor Algebras

Providing Conceptual Grounding for Contravariant Behavior

The vector space relation offers a conceptual grounding for contravariant slot behavior, moving beyond the mechanical statement of the contravariant transformation law to an explanation rooted in the natural double dual identification and its recovery of the original vector space.

Complement to the Covariant Slot's Relation to the Dual Space

The vector space relation for contravariant slots stands as the natural counterpart to the direct identification of covariant slots with elements of the dual space, together giving a complete account of how every slot of a tensor corresponds to a specific space of linear objects.