Comment by ogogmad
25 days ago
The situation is worse than I thought. The term "function", as used in foundations of mathematics, includes functionals as a special case. By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict one another.
By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict each other.
This is incorrect. In mathematics there is a single definition of function. There is no conflict or contradiction. In all cases a function is a subset of the cross product of two spaces that satisfies a certain condition.
What changes from subject to subject is what the underlying spaces of interest are.
> What changes from subject to subject is what the underlying spaces of interest are.
I'm not sure I understand what you mean here. I need some clarification. How does this have any bearing on whether functionals count as functions or not? What is the "underlying spaces of interest" in this example?
In some trivial way, every mathematical object can be seen as a function. You can replace sets in axiomatic set theory with functions.
Everything I wrote was assuming set theory as the foundations for mathematics and applies only to that setup. At any rate a functional is function since the definition starts with: a functional is a function from…
Some books will say: a functional is a linear map….
Note that a linear map is a function.
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