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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 Lemma 1,1,7 (iv) ...

Duistermaat and Kolk"s Lemma 1.1.7 reads as follows:

View attachment 7877

https://www.physicsforums.com/attachments/7878

At the start of the proof of (iv) we read the following:

" ... ... \(\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } } = \| x \|\) ... ... ... "

Suppose now we want to show, formally and rigorously that \(\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } }\)Maybe we could start with (obviously true ...)

\(\displaystyle \mid x_j \mid = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } }\) ... ... ... ... ... (1)

then we can write

\(\displaystyle \mid x_j \mid = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 }} \le ( \mid x_1 \mid^2 + \mid x_2 \mid^2 + \ ... \ ... \ + \mid x_j \mid^2 + \ ... \ ... \ + \mid x_n \mid^2 )^{ \frac{ 1 }{ 2 } } \)and we note that\(\displaystyle ( \mid x_1 \mid^2 + \mid x_2 \mid^2 + \ ... \ ... \ + \mid x_j \mid^2 + \ ... \ ... \ + \mid x_n \mid^2 )^{ \frac{ 1 }{ 2 } } = ( x_1^2 + x_2^2 + \ ... \ ... \ + x_j^2 + \ ... \ ... \ + x_n^2 )^{ \frac{ 1 }{ 2 } } = \| x \|\) ... ... ... ... (3) ... BUT ... I worry that (formally anyway) (1) is invalid ... or compromised at least ...

... for suppose for example \(\displaystyle x_j = -3\) then ...... we have LHS of (1) = \(\displaystyle \mid x_j \mid = \mid -3 \mid = 3\)... BUT ...

RHS of (1)\(\displaystyle = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } } = ( \mid -3 \mid^2 )^{ \frac{ 1 }{ 2 } } = ( 3^2 )^{ \frac{ 1 }{ 2 } } = 9^{ \frac{ 1 }{ 2 } } = \pm 3 \)My question is as follows:

How do we deal with the above situation ... and

... how do we formally and rigorously demonstrate that \(\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } } \)Hope someone can help ...

Peter

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of Lemma 1,1,7 (iv) ...

Duistermaat and Kolk"s Lemma 1.1.7 reads as follows:

View attachment 7877

https://www.physicsforums.com/attachments/7878

At the start of the proof of (iv) we read the following:

" ... ... \(\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } } = \| x \|\) ... ... ... "

Suppose now we want to show, formally and rigorously that \(\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } }\)Maybe we could start with (obviously true ...)

\(\displaystyle \mid x_j \mid = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } }\) ... ... ... ... ... (1)

then we can write

\(\displaystyle \mid x_j \mid = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 }} \le ( \mid x_1 \mid^2 + \mid x_2 \mid^2 + \ ... \ ... \ + \mid x_j \mid^2 + \ ... \ ... \ + \mid x_n \mid^2 )^{ \frac{ 1 }{ 2 } } \)and we note that\(\displaystyle ( \mid x_1 \mid^2 + \mid x_2 \mid^2 + \ ... \ ... \ + \mid x_j \mid^2 + \ ... \ ... \ + \mid x_n \mid^2 )^{ \frac{ 1 }{ 2 } } = ( x_1^2 + x_2^2 + \ ... \ ... \ + x_j^2 + \ ... \ ... \ + x_n^2 )^{ \frac{ 1 }{ 2 } } = \| x \|\) ... ... ... ... (3) ... BUT ... I worry that (formally anyway) (1) is invalid ... or compromised at least ...

... for suppose for example \(\displaystyle x_j = -3\) then ...... we have LHS of (1) = \(\displaystyle \mid x_j \mid = \mid -3 \mid = 3\)... BUT ...

RHS of (1)\(\displaystyle = ( \mid x_j \mid^2 )^{ \frac{ 1 }{ 2 } } = ( \mid -3 \mid^2 )^{ \frac{ 1 }{ 2 } } = ( 3^2 )^{ \frac{ 1 }{ 2 } } = 9^{ \frac{ 1 }{ 2 } } = \pm 3 \)My question is as follows:

How do we deal with the above situation ... and

... how do we formally and rigorously demonstrate that \(\displaystyle \mid x_j \mid \le \left( \sum_{ 1 \le j \le n } \mid x_j \mid^2 \right)^{ \frac{ 1 }{ 2 } } \)Hope someone can help ...

Peter

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