4.8.1 Tensor Scalar Codomain Case

Tensor Scalar Codomain Case is the situation in which a tensor's multilinear map produces, upon evaluation, an element of the base field $F$ itself rather than an element of some other vector space, so that the codomain of the map is simply the scalars. It is the default and most common setting in tensor algebra, encompassing ordinary bilinear forms, the metric tensor, and every fully contracted tensor, and it is the case against which the more general vector-valued codomain is defined by contrast.

Formal Definition

The Scalar-Valued Multilinear Map

A tensor in the scalar codomain case is a multilinear map

where $F$ is the same field over which the vector spaces $V_1, \ldots, V_k$ are defined. This is the setting used for the standard type $(r, s)$ tensor, whose defining map takes $r$ covectors and $s$ vectors and returns a single number.

Field as a One-Dimensional Vector Space

Formally, the scalar codomain case is the special instance of a vector-valued codomain in which the target space $W$ is taken to be $F$ itself, viewed as a one-dimensional vector space over itself. Because $F$ has dimension one, the output assignment carries no internal "direction" information beyond its numeric magnitude and sign, distinguishing this case sharply from a codomain of dimension two or higher.

Distinguishing Features of the Scalar Case

Trivial Basis for the Codomain

Since $F$ as a vector space over itself has the single basis element $1$, every scalar output is already expressed in its "component form" without any further basis choice needed on the output side. This is in contrast to a vector-valued codomain, where the output must additionally be expanded in a chosen basis of $W$ to obtain a full set of components.

Direct Correspondence With the Tensor's Own Components

In the scalar codomain case, evaluating $T$ on a full tuple of basis vectors, one per slot, produces precisely the tensor's components:

with no additional index required to label a component of the output, since the output is already a bare scalar.

Prevalence in Standard Tensor Constructions

The Metric Tensor

The metric tensor is a type $(0, 2)$ tensor with scalar codomain, taking two vectors and returning their inner product; it is the canonical example of a scalar-valued bilinear form and the structure most commonly invoked when introducing the scalar codomain case in a geometric context.

Duality Pairing

The canonical pairing between $V$ and $V^{*}$, $\langle \phi, v \rangle = \phi(v)$, is itself a scalar-valued type $(1, 1)$ tensor, and it is the elementary building block from which contraction, itself always scalar-producing when it fully saturates a tensor, is constructed.

Full Contraction of Any Tensor

Repeatedly contracting a general tensor of type $(r, s)$ against a sufficient number of vectors and covectors, one per remaining slot, always terminates in the scalar codomain case: a fully saturated tensor, with every slot filled, necessarily returns an element of $F$, regardless of how many vector-valued intermediate steps were involved along the way.

Relation to the Vector-Valued Case

Recovering Vector-Valued Maps From Scalar Ones

Every vector-valued multilinear map $T : V_1 \times \cdots \times V_k \to W$ can be recovered, without loss of information, from a family of scalar-valued maps by composing with each linear functional in a basis of $W^{}$: for a basis ${\omega_1, \ldots, \omega_m}$ of $W^{}$, the maps $\omega_j \circ T$ are each scalar-valued, and together they determine $T$ completely.

Scalar Case as the Building Block

Because of this decomposition, the scalar codomain case is often treated as the fundamental case in tensor theory, with the vector-valued case reduced to studying a finite tuple of scalar-valued tensors, one per basis functional of the target space.

Summary of Key Points

• The scalar codomain case is a multilinear map whose output lies in the base field $F$ itself, treated as a one-dimensional vector space.
• Because $F$ has a trivial one-element basis, scalar outputs require no further expansion to be read as components.
• Standard tensor objects such as the metric tensor and the canonical duality pairing are scalar-codomain examples.
• Full contraction of any tensor of any type necessarily terminates in the scalar codomain case.
• Vector-valued multilinear maps can always be decomposed into a finite family of scalar-valued maps via a basis of the codomain's dual space.