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I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ...

I need some help in order to fully understand the vector space of alternating multilinear functions ...

The relevant text from Shifrin reads as follows:

In the above text from Shifrin we read the following:

" ... ... In particular, if \(\displaystyle T \in {\bigwedge}^k ( \mathbb{R}^n )^{ \ast }\), then for any increasing \(\displaystyle k\)-tuple \(\displaystyle I\), set \(\displaystyle a_I = T( e_{ i_1} , \cdot \cdot \cdot , e_{ i_k} )\). Then we leave it to the reader to check that

\(\displaystyle T = \sum_{ i \text{ increasing } }a_I \text{dx}_I \)

... ... ... "

Can someone please help me to prove/demonstrate that \(\displaystyle T = \sum_{ i \text{ increasing } }a_I \text{dx}_I\) ... ...

Help will be much appreciated ...

Peter

==========================================================================================

In case someone needs access to the text where Shifrin defines the terms of the above post and explains the notation, I am providing access to the start of Chapter 8, Section 2.1 as follows:

Hope that helps ...

Peter

I need some help in order to fully understand the vector space of alternating multilinear functions ...

The relevant text from Shifrin reads as follows:

In the above text from Shifrin we read the following:

" ... ... In particular, if \(\displaystyle T \in {\bigwedge}^k ( \mathbb{R}^n )^{ \ast }\), then for any increasing \(\displaystyle k\)-tuple \(\displaystyle I\), set \(\displaystyle a_I = T( e_{ i_1} , \cdot \cdot \cdot , e_{ i_k} )\). Then we leave it to the reader to check that

\(\displaystyle T = \sum_{ i \text{ increasing } }a_I \text{dx}_I \)

... ... ... "

Can someone please help me to prove/demonstrate that \(\displaystyle T = \sum_{ i \text{ increasing } }a_I \text{dx}_I\) ... ...

Help will be much appreciated ...

Peter

==========================================================================================

In case someone needs access to the text where Shifrin defines the terms of the above post and explains the notation, I am providing access to the start of Chapter 8, Section 2.1 as follows:

Hope that helps ...

Peter

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