In mathematics, a real-valued function is a function whose function values are real numbers . The term real function is closely related , but it is not clearly used in the literature. Real-valued functions can be found in almost all areas of mathematics, especially in analysis , functional analysis and optimization .
A real-valued function is a function
where the target set is the set of real numbers . The definition set is arbitrary.
As with complex-valued and complex functions , the term real function is not used uniformly in the mathematical literature. Sometimes this term is synonymous with a real-valued function, sometimes only functions whose definition set is a subset of the real numbers, i.e. functions
In the case of real-valued functions, there are generally no requirements placed on the structure of the definition set. If the definition set is to be restricted, a corresponding addition is appended to the term “real-valued function”. This is the name of a function, for example
- real-valued function of a real variable, if is,
- real-valued function of several real variables, if with is,
- real-valued function of a complex variable if is,
- real-valued function of several complex variables, if with is.
If is a subset of a real vector space , then a function is also called a (real-valued) functional .
- The function is a real-valued function of a real variable.
- The function is a real-valued function of several real variables.
- The function that assigns its imaginary part to a complex number is a real-valued function of a complex variable.
- If the vector space of the symmetric real matrices , then the function is defined by a real-valued function.
- The null function is a real-valued function that is defined on any set. It assigns the number zero to each element.
The graph of a real-valued function of a real variable can be visualized by plotting the points in a two-dimensional coordinate system . To represent real-valued functions of two real variables, the points are entered in a three-dimensional coordinate system . With continuous functions, these representations form a curve or surface without jumps. With functions of two real variables, colors are sometimes used to visualize the function value. Real-valued functions of a complex variable can be represented in the same way as real-valued functions of two real variables. The imaginary part and the real part are taken as the first and second argument.
The set of all real-valued functions over a given set forms a real vector space , which is denoted by , or . The sum of two real-valued functions and is defined by
for all and the product of a real-valued function with a real number by
for everyone . These vector spaces are called real function spaces . They play an important role in linear algebra and calculus . With addition and point-wise multiplication defined by
for all the real-valued functions over the set form a commutative ring . With all three links, the real-valued functions form a real algebra .
A real-valued function is called bounded if there is a bound such that
is for everyone . The set of bounded real-valued functions forms with the supremum norm
a standardized space . Since the real numbers are complete , it is even a Banach space . A sequence of real-valued functions with for is called uniformly bounded if every term in the sequence is a bounded function and the sequence
is a bounded sequence of real numbers. A sequence of real-valued functions is called pointwise bounded if the real number sequence for all
is limited. A uniformly bounded sequence of real-valued functions is always bounded pointwise, but the converse does not have to hold. A sequence of real-valued functions is called uniformly convergent to a real-valued function if
applies. Correspondingly, a sequence of real-valued functions is called pointwise convergent to a real-valued function if for all
applies. Here, too, the uniform convergence results in point-wise convergence, but not the reverse. Further analytical properties, such as continuity , differentiability or integratability , require at least a topological , metric or mass-theoretical structure on the definition set .
After the real numbers are ordered , the partial order can be used for real-valued functions
define. A sequence of real-valued functions with is then called monotonically increasing . The partial order is analogous
defined and a sequence of real-valued functions with is then monotonically decreasing .
The real-vector- valued functions form a generalization of the real-valued functions . These are functions that are mapped in the. The vector-valued functions that map into arbitrary vector spaces are even more general . Functions that take complex function values are called complex-valued functions .
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