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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 1: Continuity ... ...
I need help with an aspect of Example 1.8.14 ... ...
The start of Duistermaat and Kolk's Example 1.8.14 reads as follows:https://www.physicsforums.com/attachments/7753In the above example we read the following:
"... ... any linear mapping $$A \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^p$$, whether bijective or not, is Lipschitz continuous and therefore uniformly continuous ... ... "I am somewhat unsure regarding proving this statement ... but I think the proof goes like the following:
For $$A \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^p$$ to be Lipschitz continuous we require
$$\mid \mid Ax - Ax' \mid \mid \le k \mid \mid x - x' \mid \mid \text{ where } x, x' \in \text{dom} (A)$$ ... ... ... (1)now we have:
$$\mid \mid Ax \mid \mid \le k \mid \mid x \mid \mid$$ ... ... ... (2)Now ... maybe put $$x = x' - x''$$
... then (2) becomes $$\mid \mid A(x' - x'') \mid \mid = \mid \mid Ax' - Ax'' \mid \mid \le k \mid \mid x' - x'' \mid \mid$$
which is the required result ...Is that correct?
Peter
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of Example 1.8.14 ... ...
The start of Duistermaat and Kolk's Example 1.8.14 reads as follows:https://www.physicsforums.com/attachments/7753In the above example we read the following:
"... ... any linear mapping $$A \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^p$$, whether bijective or not, is Lipschitz continuous and therefore uniformly continuous ... ... "I am somewhat unsure regarding proving this statement ... but I think the proof goes like the following:
For $$A \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^p$$ to be Lipschitz continuous we require
$$\mid \mid Ax - Ax' \mid \mid \le k \mid \mid x - x' \mid \mid \text{ where } x, x' \in \text{dom} (A)$$ ... ... ... (1)now we have:
$$\mid \mid Ax \mid \mid \le k \mid \mid x \mid \mid$$ ... ... ... (2)Now ... maybe put $$x = x' - x''$$
... then (2) becomes $$\mid \mid A(x' - x'') \mid \mid = \mid \mid Ax' - Ax'' \mid \mid \le k \mid \mid x' - x'' \mid \mid$$
which is the required result ...Is that correct?
Peter