Pointwise product

The pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can be multiplied (for instance, Y could be some set of numbers), then the pointwise product of f and g is another function from X to Y which maps x ∈ X to f(x)g(x).

Let X and Y be sets, and let multiplication be defined in Y—that is, for each y and z in Y let the product

be well-defined. Let f and g be functions f, g : X → Y. Then the pointwise product (f ⋅ g) : X → Y is defined by

for each x in X. In the same manner in which the binary operator ⋅ is omitted from products, we say that f ⋅ g = fg.

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.

for every real number x in R.

Let X be a set and let R be a ring. Since addition and multiplication are defined in R, we can construct an algebraic structure known as an algebra out of the functions from X to R by defining addition, multiplication, and scalar multiplication of functions to be done pointwise.

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