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

#### Peter

##### Well-known member
MHB Site Helper
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

Last edited:

#### GJA

##### Well-known member
MHB Math Scholar
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.