Definition of pointwise in mathematics?

[Mr Davis 97]

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]

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]

In this context, what does pointwise mean?

In this case "pointwise' means that the operations are defined by means of defining the result of the operation at each number ("point") in the domain of the functions. You could avoid using the term "pointwise" by correctly using the quantification "For each x". For example: if f1 and f2 are functions each having the same domain D, we define the function f = f1 + f2 as the function with domain D such that for each x in D, f(x) = f1(x) + f2(x).

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" $x$.

The usefulness of the term "pointwise" becomes clearer when you study the convergence of a sequence of functions to a limiting function. There are several distinct types of convergence, each have a different definition. "Pointwise" convergence is one type of convergence of a sequence of functions.