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I'm given a linear function $R:X\to Y$ .Then it is given $\triangle R(x)(r)=Rr.$
Can anyone please explain me what does the notation $\triangle $ $R(x)(r)$ means and denotes .. and how is it equal to $Rr$.

coool
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There is no limit or derivative involved here. When $R:\>X\to Y$ is any function then by definition $$\Delta R(x,r):=R(x+r)-R(x)\ ;$$ see also the following answer: Terminology: Delta vs... absolute?

When $R$ is linear we therefore get $$\Delta R(x,r):=R(x+r)-R\>x=R\>r\ .$$

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Christian Blatter
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If $\Delta$ is the differential, then $\Delta R$ is constant: namely, $\Delta R(x)=R$ for all $x$. See What do we mean by Derivative of linear function is a constant function..

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Martín-Blas Pérez Pinilla
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  • I know I asked the question http://math.stackexchange.com/questions/1005530/what-do-we-mean-by-derivative-of-linear-function-is-a-constant-function/1005544#1005544....why I'm asking this question is because of the following reason :the book I'm using has the proof for derivative of a linear function i.e. $R'(a)=R$ for all $a\in X$ . as follows: if $R:X\to Y$ is a linear function .Then $\triangle R(x)(r)=Rr.$ and $|\triangle R(x)(r)-Rr|=0$ for all $x,r \in X$.This shows that $D~R(x)=R$... – coool Nov 04 '14 at 11:31
  • In Short I mean to say that my proof already uses the fact $\triangle R(x)(r)=Rr.$ in proving that $R'(a)=R$ for all $a \in X$ – coool Nov 04 '14 at 11:34
  • @cool, your book proves that in fact $R′(a)=R$ for all $a$ when $R$ is linear. $R$ verifies the definition of $R'(a)$. – Martín-Blas Pérez Pinilla Nov 04 '14 at 11:36
  • can't understand what you mean by "for all $a$ when $R$ is linear."in your previous comment, can you please explain.. – coool Nov 04 '14 at 11:40
  • @coool, that in the proof (read again http://math.stackexchange.com/questions/1005530/what-do-we-mean-by-derivative-of-linear-function-is-a-constant-function/1005544#1005544) is essential that $R$ be linear. – Martín-Blas Pérez Pinilla Nov 04 '14 at 11:42
  • moreover does it mean that the proof in my book is not a proof ,it's just the verification that $R$ is that map that satisfies to be the derivative of $R'(a)$ for all $a\in X$ – coool Nov 04 '14 at 11:45
  • A verification is a very simple proof. – Martín-Blas Pérez Pinilla Nov 04 '14 at 11:49
  • sorry, but just one last doubt before accepting answer...in this verification how did we pre-define or use the fact $\triangle R(x)(r)=Rr$ ..from where does it arrive.. – coool Nov 04 '14 at 11:51
  • Now I'm in my smartphone. Later, I will edit the answer with more details. – Martín-Blas Pérez Pinilla Nov 04 '14 at 12:05
  • alright .but please do it as soon as possible for you .. – coool Nov 04 '14 at 12:07