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

I need help with another aspect of the proof of Proposition 2.3.2 ... ...

Duistermaat and Kolk's Proposition 2.3.2 and its proof read as follows:

View attachment 7850

https://www.physicsforums.com/attachments/7851

In the above proof we read the following:

" ... ... The partial differentiability of (ii) is a consequence of (i); the formula follows from \(\displaystyle Df(a) \in \text{ Lin} ( \mathbb{R}^n , \mathbb{R}^p )\) and \(\displaystyle v = \sum_{ 1 \le j \le n } v_j e_j \) ( see 1.11) ... ... "Can someone please demonstrate explicitly and rigorously how it is that the partial differentiability of (ii) is a consequence of (i) and, further, how exactly it is that the formula follows from \(\displaystyle Df(a) \in \text{ Lin} ( \mathbb{R}^n , \mathbb{R}^p )\) and \(\displaystyle v = \sum_{ 1 \le j \le n } v_j e_j\) ... ...

Help will be much appreciated ...

Peter==========================================================================================***NOTE***

It may help readers of the above post to have access to the start of Section "2.3: Directional and Partial Derivatives" ... in order to understand the context and notation of the post ... so I am providing the same ... as follows:

View attachment 7852The above post refers to (1.1) so I am providing text relevant to and including (1.1) ... as follows ...View attachment 7853Hope that the above notes/text help readers of the post understand the context and notation of the post ...

Peter