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Vector Space of Alternating Multilinear Functions ... Shifrin, Ch. 8, Section 2.1 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,886
Hobart, Tasmania
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:



Shifrin - Alternating Multilinear Functions ... .png


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:





Shifrin - 1 - Start of Ch. 8, Section 2.1 ... Differential Forms ... PART 1 ... .png
Shifrin - 2 - Start of Ch. 8, Section 2.1 ... Differential Forms ... PART 2 ... .png
Shifrin - 3 - Start of Ch. 8, Section 2.1 ... Differential Forms ... PART 3 ... .png




Hope that helps ...

Peter
 
Last edited:

GJA

Well-known member
MHB Math Scholar
Jan 16, 2013
255
Hi Peter ,

As Shifrin mentions, the $dx_{I}$ with $I$ increasing span $\Lambda^{k}(\mathbb{R}^{n})^{\ast}.$ Hence, $$T=\sum_{I\, \text{increasing}} c_{I}dx_{I}$$ for some $c_{I}$. We will show that $c_{I}=a_{I}.$

Let $J=(j_{1},\ldots, j_{k})$ be increasing. Then $$a_{J}=T(e_{j_{1}},\ldots, e_{j_{k}})=\sum_{I\, \text{increasing}}c_{I}dx_{I}(e_{j_{1}},\ldots, e_{j_{k}})=\sum_{I\,\text{increasing}} c_{I}\delta_{IJ}=c_{J},$$
as desired.