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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 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

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