Pointwise product
In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If  and  are both functions with domain  and codomain , and elements of  can be multiplied (for instance,  could be some set of numbers), then the pointwise product of  and  is another function from  to  which maps  in  to  in . Formal definition
Let  and  be sets such that  has a notion of multiplication — that is, there is a binary operation
  given by 
Then given two functions  the pointwise product  is defined by

 

for all  in . Just as we often omit the symbol for the binary operation ⋅ (i.e. we write  instead of ), we often write  for . Examples
The most common case of the pointwise product of two functions is when the codomain is a ring (or field), in which multiplication is well-defined. Algebraic application of pointwise products
Let  be a set and let  be a ring. Since addition and multiplication are defined in , we can construct an algebraic structure known as an algebra  out of the functions from  to  by defining addition, multiplication, and scalar multiplication of functions to be done pointwise. If  denotes the set of functions from  to , then we say that if  are elements of , then , , and  — the last of which is defined by
 

for all  in  — are all elements of . Generalization
If both  and  have as their domain all possible assignments of a set of discrete variables, then their pointwise product is a function whose domain is constructed by all possible assignments of the union of both sets. The value of each assignment is calculated as the product of the values of both functions given to each one the subset of the assignment that is in its domain. For example, given the function  of the boolean variables  and , and  of the boolean variables  and , both with the range in  the pointwise product of  and  is shown in the next table:

See also
Pointwise

Elementary algebra
Binary operations