scalarvalued

Scalar-valued describes quantities, maps, or functions that assign a single scalar value to each input, in contrast with vector- or tensor-valued objects. In mathematics, a scalar-valued function is one whose codomain is a field of scalars, typically the real numbers R or the complex numbers C. Such functions may be defined on sets, manifolds, or measure spaces. When a function maps into R^n or into a Banach space, it is called vector-valued (or more generally, tensor-valued).

The distinction between scalar-valued and vector-valued functions has practical implications in analysis and measure theory. A scalar-valued function is usually treated with Lebesgue integration and standard measurability concepts. Vector-valued functions often require a Bochner integral or other generalized integration theories, and their properties depend on the structure of the target space. In probability, a scalar-valued random variable is a measurable map from a probability space to R (or C), whereas a vector-valued random variable takes values in R^m.

Examples of scalar-valued functions include f(x) = x^2 on the real line, or f(z) = e^{iz} on the complex plane, as well as physical quantities like a temperature field T(p) defined on a region. In geometry and physics, scalar fields assign a single value to every point in a space, in contrast to vector fields which assign a vector.

In summary, scalar-valued emphasizes a codomain restricted to a scalar field, with broad applicability across mathematics, probability, and physics.