Real-valued function

In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

Real functions are not especially interesting in general, but many important function spaces are defined to consist of real functions.

In general

Let $X$ be an arbitrary set. Let ${\mathcal {F}}(X,{\mathbb {R} })$ to denote the set of all functions from $X$ to real numbers $R$ . Because $R$ is a field, ${\mathcal {F}}(X,{\mathbb {R} })$ is a vector space and a commutative algebra over reals:

$f+g:x\mapsto f(x)+g(x)$ – vector addition
$\mathbf {0} :x\mapsto 0$ – additive identity
$cf:x\mapsto cf(x),\quad c\in {\mathbb {R} }$ – scalar multiplication
$fg:x\mapsto f(x)g(x)$ – pointwise multiplication

Also, since $R$ is an ordered set, there is a partial order on ${\mathcal {F}}(X,{\mathbb {R} })$ :

$f\leq g\quad \iff \quad \forall x:f(x)\leq g(x)$ .

${\mathcal {F}}(X,{\mathbb {R} })$ is a partially ordered ring.

Measurable

The σ-algebra of Borel sets is an important structure on real numbers. If $X$ has its σ-algebra and a function $f$ is such that the preimage $f -1 (B)$ of any Borel set $B$ belongs to that σ-algebra, then $f$ is said to be measurable. Measurable functions also form a vector space and an algebra as explained above.

Moreover, a family of real-valued functions on $X$ can actually define a σ-algebra on $X$ generated by all preimages of all Borel sets (or of intervals only, it is not important). This is the way how σ-algebras arise in (Kolmogorov's) probability theory, where real-valued functions on the sample space $Ω$ are real-valued random variables.

Continuous

Real numbers form a topological space and a complete metric space. Continuous real-valued functions (which implies that $X$ is a topological space) are important in theories of topological spaces and of metric spaces. The concept of metric space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of continuous functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a special topological space.

Continuous functions also form a vector space and an algebra as explained above, and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) sets.

Smooth

Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be the real coordinate space (which yields a real multivariable function), a topological vector space,^[1] an open subset of them, or a smooth manifold.

Spaces of smooth functions also are vector spaces and algebras as explained above, and are a subclass of continuous functions.

Other

Other contexts where real-valued functions are used include convex functions (on vector and affine spaces), harmonic and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one or more real variables), algebraic functions (on algebraic varieties), and polynomials (of one or more real variables).

L^p spaces on sets with a measure are defined from real-valued functions too, although they are actually quotient spaces. More precisely, whereas a function satisfying an appropriate summability condition defines an element of L^p space, in the opposite direction for any $f \in L p (X)$ and $x \in X$ which is not an atom, the value $f (x)$ is undefined. Though, real-valued L^p spaces still have some of the structure explicated above. Each of L^p spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes $p$ , namely

\cdot :L^{1/\alpha }\times L^{1/\beta }\to L^{1/(\alpha +\beta )},\quad 0\leq \alpha ,\beta \leq 1.

For example, pointwise product of two L² functions belongs to L¹.

Footnotes

^ Different definition of smoothness exist in general, but for finite dimensions they are all equivalent.

External links

Weisstein, Eric W. "Real Function". MathWorld.

[1] Different definition of smoothness exist in general, but for finite dimensions they are all equivalent.

[1]