2.4 Tensor Vector Space Element Structure

Tensor Vector Space Element Structure is the description of what an individual element of a tensor space looks like, how it is represented, and what internal structure distinguishes different kinds of elements, such as simple tensors formed from a single tensor product and general tensors formed as sums of such products. It provides the bridge between the abstract characterization of a tensor space as a vector space and the concrete, workable description of the objects that populate it.

Elements as Multilinear Maps

The Intrinsic Description

Let $$V$$ be a finite-dimensional vector space over a field $$F$$. An element $$T$$ of the tensor space $$T_s^r\left(V\right)$$ is, intrinsically, a multilinear map

taking $$r$$ covectors and $$s$$ vectors as input and returning a scalar, linearly in each argument separately.

An Element as a Single Point

Regardless of how complicated the underlying multilinear map may appear, the element structure treats $$T$$ as a single, indivisible point of the vector space $$T_s^r\left(V\right)$$. It is combined with other elements using only tensor addition and scalar action, without reference to its internal multilinear formula.

Simple Tensors

Definition

An element $$T$$ is called simple, or decomposable, if it arises as a single tensor product of vectors and covectors:

where $$v_i\in V$$ and $${\omega }^j\in V^{*}$$. Evaluated on inputs, a simple tensor acts by pairing each vector or covector factor with the corresponding input argument and multiplying the resulting scalars together.

Non-Uniqueness of Decomposition

The vectors and covectors forming a simple tensor are not unique: scaling one factor by $$\alpha$$ and another by $$1/\alpha$$ leaves the element unchanged, so a simple tensor is a single point of the vector space even though many different factorizations describe it.

General Elements as Sums

Not Every Element Is Simple

For tensor spaces of type $$\left(r,s\right)$$ with $$r+s\ge 2$$, most elements are not simple. A general element is instead a finite sum of simple tensors:

shown here for type $$\left(2,0\right)$$, illustrating that a general element is built by superposing several simple tensors, exactly as tensor addition permits.

Tensor Rank of an Element

The minimal number of simple tensors required to express a given element as a sum is called its tensor rank. An element with tensor rank one is precisely a simple tensor; elements of higher rank cannot be reduced to a single tensor product, no matter how the underlying vectors and covectors are chosen.

Coordinate Description of Elements

Component Array

Fixing a basis $$e_1,\cdots ,e_n$$ of $$V$$ and its dual basis, every element $$T$$ is uniquely represented by an indexed array of scalar components

obtained by evaluating $$T$$ on all combinations of basis vectors and dual basis covectors. This array has $$n^{r+s}$$ entries and completely determines the element.

Basis Tensors as Building Blocks

The tensor products of basis vectors and dual basis covectors,

form a basis of $$T_s^r\left(V\right)$$, so that every element, simple or not, is a unique linear combination of these basis tensors with coefficients equal to its components.

Symmetry Properties of Elements

Symmetric Elements

An element $$T$$ of a purely covariant or purely contravariant type may be symmetric, meaning its value is unchanged under any permutation of its like-type arguments. Symmetric elements form a linear subspace of $$T_s^r\left(V\right)$$, since a linear combination of symmetric elements is again symmetric.

Antisymmetric Elements

Similarly, an element may be antisymmetric, changing sign under any transposition of its like-type arguments. Antisymmetric elements likewise form a linear subspace, distinct from the subspace of symmetric elements except at the zero tensor, and this distinction reflects an internal structural property specific to the individual element rather than to the tensor space as a whole.

Consequence for the Vector Space

Elements Determine the Space, Not Vice Versa

The full set of elements, encompassing every simple tensor and every finite sum of simple tensors expressible via the componentwise linear combinations, exhausts the entire tensor space; there is no element of $$T_s^r\left(V\right)$$ lying outside what can be written as such a linear combination of basis tensors, confirming that the element structure described here accounts completely for the vector space in question.