Consider a function that maps the history I-space into a space that is simpler to manage. Formally, let denote a function from a history I-space, , to a derived I-space, . The function, , is called an information mapping, or I-map. The derived I-space may be any set; hence, there is great flexibility in defining an I-map.^{11.2} Figure 11.3 illustrates the idea. The starting place is , and mappings are made to various derived I-spaces. Some generic mappings, , , and , are shown, along with some very important kinds, , and . The last two are the subjects of Sections 11.2.2 and 11.2.3, respectively. The other important I-map is , which uses the history to estimate the state; hence, the derived I-space is (see Example 11.11). In general, an I-map can even map any derived I-space to another, yielding , for any I-spaces and . Note that any composition of I-maps yields an I-map. The derived I-spaces and from Figure 11.3 are obtained via compositions.