1.2.16 Linear Functional Definition

Linear Functional Definition is the characterization of a linear functional as a linear map from a vector space to its underlying scalar field, taking a single vector as input and producing a scalar as output while respecting the operations of vector addition and scalar multiplication. It names the specific case of a linear map whose codomain is the scalar field itself, and it is precisely this notion that gives rise to the dual vector space and, through it, to the covariant side of tensor algebra.

Definition and Basic Properties

A linear functional on a vector space is a function assigning to each vector in that space a single scalar, subject to the same two conditions required of any linear map: additivity, meaning the functional applied to a sum of vectors equals the sum of the functional applied to each vector, and homogeneity, meaning the functional applied to a scaled vector equals the same scaling applied to the functional's value on that vector. Because the codomain of a linear functional is the one-dimensional scalar field itself, a linear functional can be regarded as the simplest nontrivial kind of linear map, mapping an entire vector space down to a single number in a manner compatible with its linear structure.

The expression above states the linearity condition satisfied by any linear functional, where the output on the right-hand side is guaranteed to be an ordinary scalar rather than a vector, since the functional maps into the scalar field.

Representation in Coordinates

Once a basis is chosen for a finite-dimensional vector space, every linear functional can be represented by a row of scalar coefficients, one for each basis vector, such that applying the functional to any vector amounts to multiplying its coordinate representation by this row of coefficients and summing the results. This representation makes explicit that a linear functional, once expressed in coordinates, takes the same computational form as a dot product between a fixed row of coefficients and the coordinate vector of its argument, though the functional itself is defined independently of any particular basis.

Linear Functionals and the Dual Space

The collection of every linear functional on a given vector space, together with the natural operations of adding two functionals and scaling a functional by a constant, forms a vector space in its own right: the dual space of the original vector space. This means a linear functional is not merely a device for producing numbers from vectors, but is itself a vector, belonging to the dual space, and can therefore be added to other linear functionals, scaled by constants, and combined into linear combinations exactly as ordinary vectors can. This dual identity of the linear functional — as both an operation on vectors and an element of a vector space — is what allows the dual space to inherit a full complement of algebraic structure, including its own basis, coordinates, and dimension.

Examples

The evaluation of a polynomial at a fixed point is a linear functional on the vector space of polynomials, since evaluation respects addition and scalar multiplication of polynomials. The integral of a function over a fixed interval is a linear functional on an appropriate vector space of functions, by the linearity of integration. In a finite-dimensional coordinate space, each coordinate projection — the function returning a single specified component of a vector — is itself a linear functional, and the collection of these coordinate projections forms the dual basis associated with the standard basis of that space.

Significance for Tensor Algebra

Linear functionals are the elementary building block from which the entire apparatus of covariance in tensor algebra is constructed. A covector is, by definition, an element of the dual space, which is to say, a linear functional; the natural pairing between a vector and a covector, central to the operation of contraction, is simply the act of applying a linear functional to a vector; and every covariant index carried by a tensor of higher rank corresponds, in the underlying tensor product construction, to one additional linear functional argument that the tensor, viewed as a multilinear map, is prepared to accept.