2 Vector Spaces for Tensor Algebra

Vector Spaces for Tensor Algebra is the body of prerequisite linear-algebraic material, vector spaces, bases, dual spaces, and linear maps between them, specifically framed to support the construction and manipulation of tensors, rather than covering vector space theory in its full generality. It isolates precisely the concepts a tensor construction depends on, a field of scalars, a finite-dimensional vector space over that field, its dual space of linear functionals, and the notion of a basis and its associated dual basis, before any tensor-specific machinery, contraction, index notation, or transformation laws, is introduced.

The Underlying Field and Vector Space

Scalars as the Field of Coefficients

A tensor construction begins with a choice of field F, typically the real or complex numbers, whose elements serve as coefficients throughout, and which will later appear as the values of fully contracted tensors, the type (0, 0) scalars.

The Vector Space Itself

A finite-dimensional vector space V over F is the single underlying space from which every tensor in the construction is built, either directly, as a type (1, 0) tensor is simply an element of V, or indirectly, through the dual space and their combinations.

Basis and Dimension

A basis of V is a set of n linearly independent vectors that span the space, where n, the dimension, fixes the number of independent components any type (1, 0) tensor will have once expressed in that basis.

The Dual Space

Covectors as Linear Functionals

The dual space V* consists of all linear functionals on V, maps that take a vector and return a scalar, and its elements serve directly as type (0, 1) tensors, the covectors that pair with vectors to produce numbers.

The Dual Basis

Given a basis of V, a corresponding dual basis of V* is defined by the requirement that each dual basis covector returns 1 when applied to the matching basis vector and 0 when applied to any other, a relationship that underlies the raising and lowering of indices once a metric is introduced.

Double Duality

For finite-dimensional spaces, the dual of the dual space is naturally identified with the original space itself, (V*)* ≅ V, which is what allows vectors to be regarded, symmetrically with covectors, as linear functionals acting on the dual space.

Linear Maps Between Vector Spaces

Maps as the Source of Mixed Tensors

A linear map from V to itself is naturally identified with a type (1, 1) tensor, since it can be described equivalently as a bilinear pairing that consumes a vector and a covector and returns a scalar.

Matrix Representation Relative to a Basis

Once a basis is fixed, a linear map is represented by a matrix whose entries carry exactly one upper and one lower index, matching its identification as a type (1, 1) tensor and anticipating the general rule connecting rank to the count of upper and lower indices.

Multilinear Maps as the Bridge to General Tensors

Beyond Single Linear Maps

Generalizing from a single linear map to a map that consumes several vectors and covectors simultaneously, and returns a scalar linearly in each argument separately, produces exactly the structure a general type (p, q) tensor is built to capture.

Multilinearity as the Defining Property

The defining requirement of a multilinear map, linearity in each argument separately with all other arguments held fixed, is exactly the property that allows a tensor's components, once bases for V and V* are chosen, to be organized into an array subject to the standard rules of index notation and summation.

Why These Prerequisites Matter Specifically for Tensor Algebra

Every Later Tensor Concept Rests on This Layer

Rank, type, transformation law, contraction, and symmetry are all defined in terms of the vector space V, its dual V*, chosen bases, and multilinear maps between products of these spaces, so ambiguity or weakness at this prerequisite layer propagates directly into every subsequent tensor concept built on top of it.

Isolating Only What Tensor Construction Actually Needs

This material intentionally omits much of the broader theory of vector spaces, infinite-dimensional spaces, general linear transformations unrelated to multilinearity, topics on inner product spaces beyond what is needed to introduce a metric, restricting attention to precisely the finite-dimensional, basis-dependent apparatus that tensor algebra is built directly upon.