The Dual Space and Differential Forms .... ....

[Math Amateur]

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 some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ...

The relevant text from Shifrin reads as follows:
View attachment 8791In the above text from Shifrin we read the following:

" ... ... Then \(\displaystyle \phi = a_1 dx_1 + \ ... \ ... \ + a_n dx_n\) ... ... "
Can someone please demonstrate and explain how/why \(\displaystyle \phi = a_1 dx_1 + \ ... \ ... \ + a_n dx_n\) ... ...
Help will be much appreciated ... ...

Peter

 

[GJA]

Hi Peter,

To establish the result, let $v=\sum_{k=1}^{n}v^{k}e_{k}$ be a vector in $\mathbb{R}^{n}$. Then, using linearity of $\phi$ judiciously, $$\phi(v) = \sum_{k=1}^{n}v^{k}\phi\left(e_{k}\right)=\sum_{k=1}^{n}a_{k}v^{k}.$$ On the other hand, using linearity of the $dx^{k}$,
\begin{align*}
\left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(v\right) &= \left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&= a_{1}dx^{1}\left(\sum_{k=1}^{n}v^{k}e_{k}\right) +\cdots + a_{n}dx^{n}\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&=a_{1}v^{1}+\cdots +a_{n}v^{n}.
\end{align*}
Since the two are equal and $v$ was chosen arbitrarily, we have $\phi=a_{1}dx^{1}+ \cdots + a_{n}dx^{n}.$

 

[Math Amateur]

Hi Peter,

To establish the result, let $v=\sum_{k=1}^{n}v^{k}e_{k}$ be a vector in $\mathbb{R}^{n}$. Then, using linearity of $\phi$ judiciously, $$\phi(v) = \sum_{k=1}^{n}v^{k}\phi\left(e_{k}\right)=\sum_{k=1}^{n}a_{k}v^{k}.$$ On the other hand, using linearity of the $dx^{k}$,
\begin{align*}
\left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(v\right) &= \left(a_{1}dx^{1}+\cdots + a_{n}dx^{n}\right)\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&= a_{1}dx^{1}\left(\sum_{k=1}^{n}v^{k}e_{k}\right) +\cdots + a_{n}dx^{n}\left(\sum_{k=1}^{n}v^{k}e_{k}\right)\\
&=a_{1}v^{1}+\cdots +a_{n}v^{n}.
\end{align*}
Since the two are equal and $v$ was chosen arbitrarily, we have $\phi=a_{1}dx^{1}+ \cdots + a_{n}dx^{n}.$

Thanks GJA ... most helpful ...

Appreciate your help ...

Peter