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A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2]
By definition, the map : is a relatively closed map if and only if the surjection: is a strongly closed map. If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a ...
Therefore, compilers will attempt to transform the first form into the second; this type of optimization is known as map fusion and is the functional analog of loop fusion. [2] Map functions can be and often are defined in terms of a fold such as foldr, which means one can do a map-fold fusion: foldr f z . map g is equivalent to foldr (f .
In mathematics, a linear form (also known as a linear functional, [1] a one-form, or a covector) is a linear map [nb 1] from a vector space to its field of scalars (often, the real numbers or the complex numbers).
This is an example of a non-linear functional. The Riemann integral is a linear functional on the vector space of functions defined on [a, b] that are Riemann-integrable from a to b. In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author).
The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a linear isomorphism. In the case where =, a linear map is called a linear endomorphism.
Inclusion maps are seen in algebraic topology where if is a strong deformation retract of , the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence). Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.