1.2.7 Vector Space Definition

Vector Space Definition is the precise characterization of a vector space as a set of elements called vectors, equipped with an addition operation and a scalar multiplication operation drawn from an underlying scalar field, satisfying a specific list of axioms that together guarantee the vectors can be combined and scaled in a manner consistent with the familiar behavior of directed quantities. It fixes the algebraic structure that underlies every subsequent construction in tensor algebra, since a tensor is ultimately built from one or more vector spaces and their duals.

The Axioms of a Vector Space

A vector space over a scalar field consists of a set of vectors together with two operations — vector addition, combining two vectors to produce a third, and scalar multiplication, combining a scalar with a vector to produce another vector — satisfying the following requirements: addition is associative and commutative, there exists a zero vector acting as an additive identity, every vector has an additive inverse, scalar multiplication is compatible with the field's multiplication, multiplying by the field's multiplicative identity leaves a vector unchanged, and scalar multiplication distributes over both vector addition and scalar addition.

The expression above states one of the distributive axioms a vector space must satisfy: scalar multiplication by a fixed scalar distributes over the addition of two vectors.

These axioms are stated abstractly, without reference to any particular kind of object, which is precisely what allows the notion of a vector space to apply uniformly to sets as different as arrows in physical space, sequences of numbers, polynomials, and functions, provided the chosen addition and scalar multiplication satisfy the required properties.

Examples of Vector Spaces

The most familiar example is the space of ordered tuples of real numbers of a fixed length, with addition and scalar multiplication defined componentwise. Beyond this coordinate example, the set of all polynomials of degree at most some fixed bound, with the usual addition of polynomials and multiplication by a constant, forms a vector space, as does the set of all continuous real-valued functions on an interval, or the set of all solutions to a given homogeneous linear differential equation. The generality of the vector space axioms is what allows a single body of theory — covering bases, linear independence, and linear maps — to apply uniformly across all of these otherwise dissimilar examples.

Consequences of the Axioms

From the vector space axioms alone, without any further assumption, several basic facts follow: the zero vector is unique, the additive inverse of each vector is unique, multiplying any vector by the zero scalar yields the zero vector, and multiplying the zero vector by any scalar yields the zero vector. These consequences illustrate how the axiomatic method operates within tensor algebra's foundations: a small set of stipulated properties generates, through pure deduction, a body of further facts that hold in every vector space without exception, regardless of what its elements happen to represent.

Why the Vector Space Definition Is Foundational

Every subsequent concept required to define a tensor — bases, coordinates, linear independence, linear maps, dual spaces, and multilinear maps — is defined relative to one or more vector spaces, and every one of these concepts inherits its precise meaning from the vector space axioms stated here. The tensor product itself is a construction that takes vector spaces as its input and produces a new vector space as its output, and the tensor algebra of a space is built entirely from the tensor powers of that vector space. Consequently, the vector space definition functions as the single most basic piece of algebraic apparatus in the entire foundational structure of tensor algebra, with every later definition depending on it, directly or indirectly.