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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 2: Differentiation ... ...

I need help with understanding an aspect of Definition 2.2.2 ... ...

Duistermaat and Kolk's Definition 2.2.2 reads as follows:https://www.physicsforums.com/attachments/7789

https://www.physicsforums.com/attachments/7790Towards the end of the above definition, D&K write the following:

" ... ... In the case where \(\displaystyle n = p = 1\), the mapping \(\displaystyle L \mapsto L(1)\) gives a linear isomorphism \(\displaystyle \text{End} ( \mathbb{R} ) \ \ \tilde{ \rightarrow } \ \ \mathbb{R}\). ... ... "I do not understand the above remark ... never mind why it is true ... can someone please explain what D&K mean and why it is true ...

Peter=========================================================================================I think D&K's preceding notes on Differentiable Mappings may be helpful to MHB members trying the understand the above post ... so I am providing the same as follows ... ...

View attachment 7791

View attachment 7792

I also think D&K's preceding notes on Linear Mappings may be helpful to MHB members trying the understand the above post ... so I am providing the same as follows ... ... https://www.physicsforums.com/attachments/7793

View attachment 7794

View attachment 7795Hope the above text helps ...

Peter

I am focused on Chapter 2: Differentiation ... ...

I need help with understanding an aspect of Definition 2.2.2 ... ...

Duistermaat and Kolk's Definition 2.2.2 reads as follows:https://www.physicsforums.com/attachments/7789

https://www.physicsforums.com/attachments/7790Towards the end of the above definition, D&K write the following:

" ... ... In the case where \(\displaystyle n = p = 1\), the mapping \(\displaystyle L \mapsto L(1)\) gives a linear isomorphism \(\displaystyle \text{End} ( \mathbb{R} ) \ \ \tilde{ \rightarrow } \ \ \mathbb{R}\). ... ... "I do not understand the above remark ... never mind why it is true ... can someone please explain what D&K mean and why it is true ...

Peter=========================================================================================I think D&K's preceding notes on Differentiable Mappings may be helpful to MHB members trying the understand the above post ... so I am providing the same as follows ... ...

View attachment 7791

View attachment 7792

I also think D&K's preceding notes on Linear Mappings may be helpful to MHB members trying the understand the above post ... so I am providing the same as follows ... ... https://www.physicsforums.com/attachments/7793

View attachment 7794

View attachment 7795Hope the above text helps ...

Peter

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