Reellwertiger

Reellwertiger is a term used in German mathematics to describe objects whose values lie in the real numbers, as opposed to complex- or vector-valued ones. It is commonly applied to functions, random variables, and sequences, indicating that every output is a real number.

A reellwertige Funktion f maps from some domain X into the real numbers, written as f: X → ℝ. A real-valued random variable is a measurable function from a probability space to ℝ. Likewise, a real-valued sequence (a_n) has each term a_n ∈ ℝ. The designation emphasizes the absence of imaginary or other non-real values.

In some mathematical settings, a distinction is made between reellwertig and erweiterter reellwertig. Extended real-valued functions take values in the extended real line [-∞, ∞], which is common in contexts involving limits, infima, or improper integrals. Real-valued functions, by contrast, have outputs strictly in ℝ.

The concept is often contrasted with komplexwertig (complex-valued) or komplex-valuige mappings. A function f: X → ℂ is real-valued precisely when Im f(x) = 0 for all x ∈ X, in which case it is also considered reellwertig.

In analysis and measure theory, real-valued functions enjoy order structure, allowing notions such as nonnegativity (f ≥ 0) and monotonicity. Many standard theorems and definitions (continuity, integrability, measurability) are stated for real-valued objects, though they often extend to complex-valued or extended real-valued cases as needed.