Real-valued function
Algebraic structure

Let F (X, R) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} be the set of all functions from a set X to real numbers R {\displaystyle \mathbb {R} }. Because R {\displaystyle \mathbb {R} } is a field, F (X, R) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} may be turned into a vector space and a commutative algebra over the reals with the following operations:

These operations extend to partial functions from X to R, {\displaystyle \mathbb {R},} with the restriction that the partial functions f + g and f g are defined only if the domains of f and g have a nonempty intersection; in this case, their domain is the intersection of the domains of f and g.

Also, since R {\displaystyle \mathbb {R} } is an ordered set, there is a partial order

on F (X, R), {\displaystyle {\mathcal {F}}(X,{\mathbb {R} }),} which makes F (X, R) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} a partially ordered ring.
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