4.9.2 Tensor Scalar Valued Form Relation

Tensor Scalar Valued Form Relation is the correspondence linking the general notion of a scalar-valued multilinear map to the more specialized notion of a differential form, which is precisely a scalar-valued multilinear map that is additionally required to be totally antisymmetric under permutation of its arguments. It clarifies that every form is a scalar-valued multilinear map, but the reverse does not hold: forms are the antisymmetric subclass singled out within the much larger space of all scalar-valued multilinear maps of a given order.

Formal Definition

Forms as a Restricted Subclass

A scalar-valued multilinear map $\omega : \underbrace{V \times \cdots \times V}_{k} \to F$ is called a $k$-form (or an alternating $k$-linear form) precisely when it satisfies the additional antisymmetry condition

for every permutation $\sigma$ of ${1, \ldots, k}$. The relation between forms and the general class of scalar-valued multilinear maps is therefore an inclusion: the space $\Lambda^{k}(V^{*})$ of $k$-forms is a linear subspace of the full space $T^{0}_{k}(V)$ of scalar-valued type $(0,k)$ multilinear maps, cut out by the antisymmetry constraint.

Forms Require a Repeated Vector Space Domain

Because antisymmetry compares the map's value under exchange of two arguments, the notion of a form only makes sense when every slot draws from the same space, i.e., on a repeated vector space domain of the type $V^{k}$; a scalar-valued multilinear map on a domain built from genuinely distinct factor spaces cannot be classified as a form, since there is no well-typed way to permute arguments across unrelated spaces.

Consequences of the Relation

Dimension Comparison

The space of all scalar-valued type $(0,k)$ multilinear maps on an $n$-dimensional $V$ has dimension $n^{k}$, while the subspace of $k$-forms has the strictly smaller dimension

for $k \leq n$, and dimension zero for $k > n$. This sharp reduction in dimension is a direct numerical illustration of how restrictive the antisymmetry condition is relative to the unconstrained scalar-valued multilinear map structure.

Every Form Inherits Slotwise Linearity

Because a $k$-form is defined as a special case of a scalar-valued multilinear map, it automatically inherits the full slotwise linearity property in every argument, without needing to verify additivity and homogeneity separately; only the extra antisymmetry condition needs to be checked once slotwise linearity is already assumed, since antisymmetry presupposes, but does not replace, multilinearity.

Vanishing on Repeated Arguments

A direct consequence of total antisymmetry, available only because the underlying object is a scalar-valued multilinear map to begin with, is that a $k$-form vanishes whenever any two of its arguments coincide, since swapping two equal arguments must simultaneously leave the value unchanged (equal arguments) and reverse its sign (antisymmetry), forcing the value to be zero.

Extracting Forms From General Maps

Alternation as a Projection

Given an arbitrary scalar-valued multilinear map $T$ of type $(0,k)$, applying the alternation operator

produces a genuine $k$-form, and this operator acts as a linear projection from the full space of scalar-valued multilinear maps onto the subspace of forms, showing explicitly how any general scalar-valued map relates to, and can be converted into, an antisymmetric one.

Summary of Key Points

• A form is precisely a scalar-valued multilinear map that additionally satisfies total antisymmetry under permutation of its arguments.
• Forms constitute a proper linear subspace of the space of all scalar-valued multilinear maps of the same order, cut out by the antisymmetry condition.
• Forms require a repeated vector space domain, since antisymmetry is only well-typed when all slots draw from the same underlying space.
• The dimension of the space of $k$-forms, $\binom{n}{k}$, is strictly smaller than the dimension $n^{k}$ of the unconstrained space, illustrating the restrictiveness of antisymmetry.
• The alternation operator provides an explicit linear projection converting any scalar-valued multilinear map into a genuine form.