FUNCTIONAL ANALYSIS OF MAPPING

ABSTRACT

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function’s "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x² by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x²). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

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