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- Thread starter Mr Davis 97
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mathman

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

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Mark44

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That would be "scalwhere a is a scaler

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Svein

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"operations on real functions in a vector space are defined pointwise, such thatf1+f2=f1(x)+f2(x)f_1 + f_2 = f_1(x) + f_2(x), andaf=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

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Stephen Tashi

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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 [itex] f [/itex] and [itex] g [/itex] whose domain is the real numbers we can define [itex] H = f\circ g [/itex] to be the function [itex] H(x) = \int_{-\infty}^{\infty} f(y)g(x-y) dy [/itex] (provided this definite integral exists). The integration in the definition is not a "pointwise" operation on [itex] f [/itex] and [itex] g [/itex] since the outcome depends on their values at more than one "point". However, one can say that [itex] H [/itex] is defined "pointwise" in the sense that its definition explains [itex] H [/itex] by telling its value at each "point" [itex] x[/itex].

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.

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