4.9 Tensor Scalar Valued Multilinear Map Structure

Tensor Scalar Valued Multilinear Map Structure is the complete algebraic framework describing a tensor as a map that takes several vector and covector arguments and returns a single element of the base field, combining a slotwise-linear domain structure with a scalar codomain into one coherent object. It is the most fundamental and widely used form of tensor, encompassing bilinear forms, the metric, contraction pairings, and every tensor whose ultimate purpose is to produce a number from a collection of vectors.

Formal Definition

The Full Structure

A scalar-valued multilinear map of type $(r, s)$ over a vector space $V$ is a function

satisfying the slotwise linearity property in every one of its $r+s$ argument slots. The structure is complete once three pieces of data are fixed: the domain (an ordered, variance-labeled product of copies of $V$ and $V^{*}$), the linearity condition holding independently in each slot, and the scalar codomain $F$.

Space of All Such Maps

The collection of all scalar-valued multilinear maps of a fixed type $(r, s)$ on $V$ forms a vector space in its own right, denoted $T^{r}{s}(V)$, under pointwise addition and scalar multiplication of maps. This space is naturally isomorphic to the tensor product $\underbrace{V \otimes \cdots \otimes V}{r} \otimes \underbrace{V^{} \otimes \cdots \otimes V^{}}{s}$, which is the origin of calling elements of $T^{r}{s}(V)$ "tensors of type $(r,s)$."

Structural Features

Components as the Concrete Realization

Once a basis ${e_1, \ldots, e_n}$ of $V$ is fixed, along with the dual basis ${e^1, \ldots, e^n}$, the structure reduces to a finite array of scalar components

with $n^{r+s}$ entries in total, and every evaluation of $T$ on arbitrary arguments reduces to a finite weighted sum over these components, with weights given by products of coordinates of the arguments.

Transformation Law Under Change of Basis

Under a change of basis of $V$ given by an invertible matrix $A$, the components transform with $r$ factors of $A^{-1}$ and $s$ factors of $A$, reflecting the mixed variance of the domain structure. This transformation rule is what makes the scalar-valued multilinear map structure a genuine tensor, as opposed to a basis-dependent array of numbers with no coordinate-invariant meaning.

Common Instances

The Metric Tensor as Type (0,2)

The metric tensor is a scalar-valued multilinear map of type $(0, 2)$, taking two vectors and returning their inner product; it is additionally required to be symmetric and (in the Riemannian case) positive-definite, constraints layered on top of the basic scalar-valued multilinear structure.

The Canonical Pairing as Type (1,1)

The pairing $\langle \phi, v \rangle = \phi(v)$ between $V^{*}$ and $V$ is the simplest genuinely mixed scalar-valued multilinear map, of type $(1,1)$, and it underlies the definition of contraction for every higher-type tensor.

Determinant-Like Maps

The determinant, viewed as a function of $n$ vectors in an $n$-dimensional space, is a scalar-valued multilinear map of type $(0, n)$ that is additionally totally antisymmetric, illustrating how symmetry constraints can be imposed on top of the basic scalar-valued structure to produce specialized tensors like volume forms.

Operations Native to This Structure

Full Contraction to a Number

Because the output is already a scalar, no further contraction is needed once every slot is filled; scalar-valued multilinear maps are, in this sense, the terminal objects of the contraction process applied to any higher-type tensor.

Tensor Product of Two Scalar-Valued Maps

Given scalar-valued maps $S$ of type $(r_1, s_1)$ and $T$ of type $(r_2, s_2)$, their tensor product $S \otimes T$, defined by evaluating each on its own separate block of arguments and multiplying the two scalar results, is again a scalar-valued multilinear map, now of type $(r_1+r_2,, s_1+s_2)$.

Summary of Key Points

• A scalar-valued multilinear map structure combines a variance-labeled domain of $V$ and $V^{*}$ factors with slotwise linearity and a scalar codomain.
• The collection of all such maps of a fixed type forms a vector space isomorphic to the corresponding tensor product space.
• Fixing a basis reduces the structure to a finite component array of size $n^{r+s}$, transforming with $r$ copies of $A^{-1}$ and $s$ copies of $A$.
• Familiar tensors such as the metric, the canonical pairing, and the determinant are all instances of this structure, often with additional symmetry constraints layered on top.
• The tensor product of two scalar-valued maps, multiplying their separately evaluated outputs, produces a new scalar-valued map of combined type.