What does the term "pointwise" refer to?

[Mr Davis 97]

In reading mathematical texts (especially on functions), I come across the term "pointwise". There seems to be a paucity of information explaining it on the internet, so I turn to the forum for an intuitive explanation. The context is when the term is used as in "pointwise addition" or "pointwise multiplication" of functions.

 

[SteamKing]

In reading mathematical texts (especially on functions), I come across the term "pointwise". There seems to be a paucity of information explaining it on the internet, so I turn to the forum for an intuitive explanation. The context is when the term is used as in "pointwise addition" or "pointwise multiplication" of functions.

This often refers to a graphical method of adding or multiplying the ordinates of two curves to obtain a third curve, which is the sum or product of the first two curves.

Examine:

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You take a series of points from curves ch.1 and ch.2 at the same x-location and add them together algebraically to make the bottom curve in the graphic above.

 

[mathman]

Let f,g,h be functions. f+g=h means f(x)+g(x)=h(x) for every x in the domain of these function. Similarly for any other operation involving functions. The term is more often used in the context of limits, where $f_n -> f$ can be defined in different ways, one of which is pointwise.

 

[Mr Davis 97]

So pointwise basically means that the way we manipulate functions is by considering their values at any x?

 

[Mr Davis 97]

Let f,g,h be functions. f+g=h means f(x)+g(x)=h(x) for every x in the domain of these function. Similarly for any other operation involving functions. The term is more often used in the context of limits, where $f_n -> f$ can be defined in different ways, one of which is pointwise.

Bump...

 

[Mark44]

So pointwise basically means that the way we manipulate functions is by considering their values at any x?

No, at a specific x value.

"Pointwise" is commonly used in discussions about the convergence of sequences. See https://en.wikipedia.org/wiki/Pointwise_convergence for more info. One example given in this wiki article is the sequence $\{x^n\}$ on the interval [0, 1). This sequence converges pointwise to 0 for each x in [0, 1), but does not converge uniformly to 0 on the same interval.