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enter image description here I have 4 questions about this:

1)What is meant by $\mathbb{R}^2 \rightarrow \mathbb{R}$ in this context?

2)What is meant by $C=kA+(1−k)B$?Aren't $A,B,C$ just points in euclidean plane?How can they be related by an equation?

3)Why is it sufficient to show the condition for points $A$ and $B$?

4)What is meant by $F$ is linear?How to prove it and what is it's use?

Sorry if that's too obvious.I am not aware of the terminology used here i guess.

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1)What is meant by R2→R in this context?

$F$ is a function whose domain is $\mathbb{R}^2$ and codomain is $\mathbb{R}$. The domain is $\mathbb{R}^2$ because its inputs are points, and the codomain is $\mathbb{R}$ because its outputs are real numbers. Since what $F$ does is ignore the $\omega$ terms in $\mathbb{P}$, it is what is called a `partial function' in programming. Put another way, the domain of $\mathbb{P}$ is $\mathbb{R}^2 \times \mbox{the space of circles}^2$ (sorry, I don't know what space $\omega$ lives in), and its codomain is $\mathbb{R}$. What $F$ does is ignore some of the inputs.

2)What is meant by C=kA+(1−k)B?Aren't A,B,C just points in euclidean plane?How can they be related by an equation?

It's a statement that $C$ is on the line segment between $A$ and $B$. This kind of formula: $kx + (1-k)y$ is called a convex combination of $x$ and $y$. In the context of two points in Euclidean space, it is a way to associate the points on the line thru $x$ and $y$ with values of $k$. If $0 \leq k\leq 1$, then the point is between $x$ and $y$.

3)Why is it sufficient to show the condition for points A and B?

4)What is meant by F is linear?How to prove it and what is it's use?

The answers to 3 and 4 are connected. A function $f$ is linear if $f(a + cb) = f(a) + c f(b)$ no matter what $a$ and $b$ (and $c$) are. So the statement about sufficiency is just a bit of semantic fluff to get the proof started. Any time you want to prove a function is linear, you start with a statement similar to: "Let $a$ and $b$ be in the domain of the function".

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  • $\begingroup$ Thanks for the help but in the 1st point i know the domains and codomains are $\mathbb{R}^2$ and $\mathbb{R}$.But what does it mean for $\mathbb{R}^2$ and $\mathbb{R}$ here?I mean what role does $P$ play as $\mathbb{R}^2$ and $\mathbb{P}(P, \omega)$ as $\mathbb{R}$? $\endgroup$
    – a_i_r
    Mar 19, 2021 at 13:48
  • $\begingroup$ OK, I think I understand your question. I'll edit my answer. $\endgroup$ Mar 19, 2021 at 13:54
  • $\begingroup$ I don't think I'd say $F$ "ignores" the circles. Rather, to get $F$ we first have to fix the choice of the two circles so that the only parameter of $\mathbb P$ that is allowed to vary is $P$. Moreover, I would call $F$ a partially applied function. A partial function is one that does not produce output values for every value in its domain. $\endgroup$
    – David K
    Mar 19, 2021 at 14:17
  • $\begingroup$ Thanks again but i have one last inquiry.$f$ is linear if $f(a+cb)=f(a)+cf(b)$.What are $a,b,c$ according to the context?I mean the proof mentioned in the post? $\endgroup$
    – a_i_r
    Mar 19, 2021 at 14:30
  • $\begingroup$ @DavidK in programming this is indeed what's called a partial function. See for example: docs.python.org/3/library/functools.html. As for the ignores / fixes distinction, yes I agree that's also a fine way to think about it. $\endgroup$ Mar 19, 2021 at 19:32

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