vectorvalued

Vector-valued describes mathematical objects whose values lie in a vector space rather than in a field of scalars. In practice, vector-valued data take the form of vectors, points, or more generally elements of a vector space over R or C. This contrasts with scalar-valued objects, whose outputs are real or complex numbers.

A common instance is a vector-valued function f: X → V, where X is a set or space and V is a vector space. When V = R^n, f is often described by its coordinate functions f1, …, fn: X → R. Examples include a position function t ↦ (x(t), y(t), z(t)) in space or a signal t ↦ (f1(t), …, fn(t)) with multiple channels.

Topological and analytic properties generalize componentwise in finite-dimensional V; for infinite-dimensional V one uses a norm or inner product. Continuity, differentiability, and integrability extend from real-valued functions to vector-valued ones in standard ways.

Integration and measurement: vector-valued functions can be integrated in the Bochner sense, yielding elements of V. The associated function spaces include L^p(X; V). We distinguish strong (Bochner) integrals from weak (Pettis) integrals.

Applications: vector-valued maps arise in geometry (maps between manifolds and vector fields), physics (velocity and force fields), computer graphics (color channels as vectors), and statistics (random vectors). The concept emphasizes that the output resides in a linear space, enabling linear operations and analysis.