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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 1.5.3 ...

Duistermaat and Kolk"s Example 1.5.3 reads as follows:View attachment 7707In the above example we read the following:

" ... ... Then the orthogonal projection \(\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}^p\) with \(\displaystyle f(x) = ( x_1, \ ... \ ... x_p )\) ... ... "My question regards D&K's understanding of an orthogonal projection ... ...Wikipedia describes a projection (orthogonal?) as follows:

" In linear algebra and functional analysis, a

that

Indeed can someone please explain the nature of an orthogonal projection and how D&K's function f qualifies as such ... ...Help will be much appreciated ...

Peter

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

I need help with an aspect of Example 1.5.3 ...

Duistermaat and Kolk"s Example 1.5.3 reads as follows:View attachment 7707In the above example we read the following:

" ... ... Then the orthogonal projection \(\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R}^p\) with \(\displaystyle f(x) = ( x_1, \ ... \ ... x_p )\) ... ... "My question regards D&K's understanding of an orthogonal projection ... ...Wikipedia describes a projection (orthogonal?) as follows:

" In linear algebra and functional analysis, a

**projection**is a linear transformation*P*from a vector space to itself suchthat

*P*^{2}=*P*. That is, whenever*P*is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged."How do we square D&K's orthogonal projection f with Wikipedia's definition of a projection ... ?Indeed can someone please explain the nature of an orthogonal projection and how D&K's function f qualifies as such ... ...Help will be much appreciated ...

Peter

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