Reallinear

Reallinear, more commonly written as real-linear, refers to a property of a map between real vector spaces that is linear with respect to real scalars. A real-linear map preserves both vector addition and scalar multiplication by real numbers, distinguishing it from linear maps over other fields such as the complex numbers.

Formally, let V and W be real vector spaces. A function T: V → W is real-linear if for all u, v in V and all real numbers a, b, T(a u + b v) = a T(u) + b T(v). This condition is equivalent to T being additive (T(u+v) = T(u) + T(v)) and homogeneous with respect to real scalars (T(a v) = a T(v) for all real a). The collection of all real-linear maps from V to W is denoted Hom_R(V, W) or L_R(V, W).

Examples include the identity map on R^n, and any matrix multiplication by a fixed real matrix A, which defines a real-linear map from R^n to R^m. The differentiation operator on suitable spaces of real-valued functions is another real-linear operator. It is important to note that a real-linear map is not necessarily complex-linear; preserving real scalars does not imply preserving multiplication by complex numbers.

Properties include that real-linear maps form a real vector space, and when V and W are finite-dimensional, such maps correspond to matrices with real entries. Composition of real-linear maps is real-linear, and the set Hom_R(V, W) can be studied with standard linear-algebra tools.

Etymology and usage: the standard term is real-linear. The unhyphenated form “reallinear” is uncommon in mathematical literature and may appear only in informal contexts or branding.