# Definition of pointwise in mathematics?

## Main Question or Discussion Point

I have tried to search on the internet for a clear and concise definition for the mathematical term "pointwise," but I cannot find one that is comprehensible. The context of needing an answer to this question is this: "operations on real functions in a vector space are defined pointwise, such that $f_1 + f_2 = f_1(x) + f_2(x)$, and $af = a(f(x))$, where a is a scaler." In this context, what does pointwise mean?

mathman
It means that the operations are valid for every x in the domain of the functions.

Mark44
Mentor
where a is a scaler
That would be "scalar."

Svein
"operations on real functions in a vector space are defined pointwise, such that f1+f2=f1(x)+f2(x)f_1 + f_2 = f_1(x) + f_2(x), and af=a(f(x))af = a(f(x)), where a is a scalar." In this context, what does pointwise mean?
You are talking about operations on functions, not numbers.We could define operations on functions without referring to the numbers (points) they are operating on, but the easiest way is to define the value of a sum of two functions as the sum of the values of the two functions. Likewise, the easiest way of defining the value of a constant times a function is to define it as the constant times the value of the function. That is what we call defining it pointwise.

Stephen Tashi
As an example of an operation on functions that isn't "pointwise", we can consider the operation of convolution of two functions. For functions $f$ and $g$ whose domain is the real numbers we can define $H = f\circ g$ to be the function $H(x) = \int_{-\infty}^{\infty} f(y)g(x-y) dy$ (provided this definite integral exists). The integration in the definition is not a "pointwise" operation on $f$ and $g$ since the outcome depends on their values at more than one "point". However, one can say that $H$ is defined "pointwise" in the sense that its definition explains $H$ by telling its value at each "point" $x$.