4.5.2 Tensor Argument Slot Vector Space

Tensor Argument Slot Vector Space is the specific vector space, either the base space $V$ or its dual $V^{*}$, that a given argument slot of a tensor's multilinear map is assigned to draw its input from. Every slot in a tensor's argument slot structure is not an abstract placeholder alone; it is bound to a concrete vector space that determines what kind of object, and from what set, may legally be substituted into that position.

Formal Definition

Binding a Slot to a Space

For a tensor represented as a multilinear map with argument slot structure of $r$ contravariant and $s$ covariant positions,

each of the $r$ contravariant slots is bound to the dual space $V^{}$, and each of the $s$ covariant slots is bound to $V$ itself. The slot vector space is precisely this binding: a function that maps each position in the argument slot set to one of $V$ or $V^{}$.

Consistency Requirement

For the map $T$ to be well-defined and multilinear, every substitution into a given slot must come from that slot's designated vector space. Substituting a covector where a vector is expected, or vice versa, is not merely disallowed by convention; it is undefined because the linear structure being invoked (addition, scalar multiplication) in that slot only exists relative to the specific space assigned to it.

The Duality Between V and V*

Why Two Spaces Are Needed

A single vector space $V$ alone does not supply enough structure to build tensors of mixed type, because a purely covariant tensor and a purely contravariant tensor behave differently under change of basis. Introducing the dual space $V^{}$, whose elements are linear functionals on $V$, gives a second, dimensionally-matched space with an inverse transformation law: if $V$ has dimension $n$, so does $V^{}$, and there is a canonical pairing

between them. Each contravariant slot of a tensor draws from $V^{*}$ precisely because that pairing is what allows the slot, when filled, to consume a vector's worth of information and return a scalar contribution.

Basis and Dual Basis Per Slot

If ${e_1, \ldots, e_n}$ is a basis of $V$, the corresponding dual basis ${e^1, \ldots, e^n}$ of $V^{*}$ satisfies

where $\delta^{i}_{j}$ is the Kronecker delta. Every contravariant slot, when expanded in this dual basis, and every covariant slot, when expanded in the primal basis, together supply the full component expansion of the tensor described elsewhere for the domain product.

Effects on Tensor Operations

Index Raising and Lowering

When a metric tensor $g$ is available, it provides a canonical isomorphism between $V$ and $V^{}$, which allows the vector space bound to a given slot to be changed: a covariant slot's space $V$ can be swapped for $V^{}$ (raising the index), or a contravariant slot's space $V^{*}$ can be swapped for $V$ (lowering the index), while the total arity of the tensor stays the same.

Slot Space and Contraction

Contraction is only defined between a slot bound to $V^{}$ and a slot bound to $V$, because contraction is literally the evaluation of the canonical pairing between those two spaces. Two slots bound to the same space (both $V$ or both $V^{}$) cannot be contracted directly unless an additional structure, such as a metric, is introduced to identify the two copies of $V$ (or $V^{*}$) with each other.

Summary of Key Points

• Every argument slot of a tensor is bound to a specific vector space, either $V$ or its dual $V^{*}$.
• Contravariant slots draw from $V^{*}$ and covariant slots draw from $V$, and this binding determines the tensor's type $(r,s)$.
• The canonical pairing between $V$ and $V^{*}$ is what allows contraction between a matched pair of slots to produce a scalar.
• A metric tensor provides an isomorphism between $V$ and $V^{*}$, enabling the vector space bound to a slot to be changed via index raising or lowering.
• Contraction requires one slot bound to $V^{*}$ and one bound to $V$; same-space slot pairs require extra structure to be paired meaningfully.