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Don’t know if you are a mathematician or not but mathematically speaking “function” has a definition that is valid in all mathematical contexts. Functional cle

Don’t know if you are a mathematician or not but mathematically speaking “function” has a definition that is valid in all mathematical contexts. Functional clearly meets the criteria to be a function since being a function is part of the definition of being a functional.

ogogmad 2y 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.
- skhunted 2y ago
  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.
  - ogogmad 2y ago
    > 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.
    - skhunted 2y ago
      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.
      - ogogmad 2y ago
        You genuinely don't know what you're talking about. The word "function" means different things in different areas. So does the word "map" or "mapping". In analysis, what you personally call a "function" instead falls under the term "mapping". In foundations - which is a different area with incompatible terminology - the terms "mapping" and "function" are defined to mean the same thing. This situation is a consequence of how mathematicians haven't always been sure how to define certain concepts. See "generating function" for yet another usage of the word "function" that's in direct contradiction with the last two. So that's three incompatible usages of the term "function". All this terminology goes back to the 1700s when mathematics was done without the rigour it has today. I find it aggravating how you're so confidently wrong. I hope it's not on purpose. [edit] [edit 2: Removed insults]
        skhunted 2y ago
        You genuinely don't know what you're talking about. .... I find it aggravating how you're so confidently wrong. This is a fine example of irony. Let V be a vector space over the reals and L a functional. Let v be a particular element of V. L(v) is a real number. It is a single value. L(v) can't be 1.2 and also 3.4. Thus L is a function. A function is simply a subset of the product of two sets with the property that if (a,b) and (a, c) are in this subset then b=c. Can you find a functional that does not meet this criterion? If so then you have an object such that L maps v to a and also maps v to c with a and c being different elements. Find me a linear map that does not meet the definition of function. Give an example of a functional in which the functional takes a given input to more than one element of the target set. I think you are not a mathematician and you also don't appear to understand that a word can have different meanings based on context. "generating function" isn't the same thing as "function". Notice that generating is paired with function in the first phrase. Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!

ogogmad 2y 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.

skhunted 2y ago
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.
- ogogmad 2y ago
  > 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.
  - skhunted 2y ago
    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.
    - ogogmad 2y ago
      You genuinely don't know what you're talking about. The word "function" means different things in different areas. So does the word "map" or "mapping". In analysis, what you personally call a "function" instead falls under the term "mapping". In foundations - which is a different area with incompatible terminology - the terms "mapping" and "function" are defined to mean the same thing. This situation is a consequence of how mathematicians haven't always been sure how to define certain concepts. See "generating function" for yet another usage of the word "function" that's in direct contradiction with the last two. So that's three incompatible usages of the term "function". All this terminology goes back to the 1700s when mathematics was done without the rigour it has today. I find it aggravating how you're so confidently wrong. I hope it's not on purpose. [edit] [edit 2: Removed insults]
      - skhunted 2y ago
        You genuinely don't know what you're talking about. .... I find it aggravating how you're so confidently wrong. This is a fine example of irony. Let V be a vector space over the reals and L a functional. Let v be a particular element of V. L(v) is a real number. It is a single value. L(v) can't be 1.2 and also 3.4. Thus L is a function. A function is simply a subset of the product of two sets with the property that if (a,b) and (a, c) are in this subset then b=c. Can you find a functional that does not meet this criterion? If so then you have an object such that L maps v to a and also maps v to c with a and c being different elements. Find me a linear map that does not meet the definition of function. Give an example of a functional in which the functional takes a given input to more than one element of the target set. I think you are not a mathematician and you also don't appear to understand that a word can have different meanings based on context. "generating function" isn't the same thing as "function". Notice that generating is paired with function in the first phrase. Example: Jellyfish is not a jelly and not a fish. Biologists have got it all wrong!