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I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ...

I need help in order to fully understand Tu's section on the dual space ... ... In his section on the dual space, Tu writes the following:

View attachment 8792In the above text from Tu, just preliminary to Proposition 3.1 Tu writes the following:

" ... ... Let \(\displaystyle e_1, \ ... \ ... \ e_n\) be a basis for \(\displaystyle V\). ... ... "

Now my question is as follows:Is \(\displaystyle e_1, \ ... \ ... \ e_n\) a completely general basis ... hence making Proposition 3.1 completely general (in terms of basis anyway) ... or given the notation is \(\displaystyle e_1, \ ... \ ... \ e_n\) the standard basis where \(\displaystyle e_1 = ( 1, 0, 0, \ ... \ ... \ , 0 )^T , e_2 = ( 0,1, 0, \ ... \ ... \ , 0 )^T , \ ... \ ... \ , \ e_n = ( 0, 0, 0, \ ... \ ... \ , 1 )^T\) ...I must say I can find no instance in the proof of Proposition 3.1 where the basis \(\displaystyle e_1, \ ... \ ... \ e_n\) is assumed to be anything but completely general ... but would be appreciative if someone would confirm this to be the case ... ... ( ... hmmm ... pity Tu didn't use \(\displaystyle u_1, u_2, \ ... \ ... \ , u_n\) as the basis for \(\displaystyle V\) ... ... )My suspicions about the basis being not general ... but indeed the standard basis ... ... came about on reading the following note on page 11 concerning notation ...

View attachment 8793Hope someone can clarify the above ...

Help will be appreciated ...

I need help in order to fully understand Tu's section on the dual space ... ... In his section on the dual space, Tu writes the following:

View attachment 8792In the above text from Tu, just preliminary to Proposition 3.1 Tu writes the following:

" ... ... Let \(\displaystyle e_1, \ ... \ ... \ e_n\) be a basis for \(\displaystyle V\). ... ... "

Now my question is as follows:Is \(\displaystyle e_1, \ ... \ ... \ e_n\) a completely general basis ... hence making Proposition 3.1 completely general (in terms of basis anyway) ... or given the notation is \(\displaystyle e_1, \ ... \ ... \ e_n\) the standard basis where \(\displaystyle e_1 = ( 1, 0, 0, \ ... \ ... \ , 0 )^T , e_2 = ( 0,1, 0, \ ... \ ... \ , 0 )^T , \ ... \ ... \ , \ e_n = ( 0, 0, 0, \ ... \ ... \ , 1 )^T\) ...I must say I can find no instance in the proof of Proposition 3.1 where the basis \(\displaystyle e_1, \ ... \ ... \ e_n\) is assumed to be anything but completely general ... but would be appreciative if someone would confirm this to be the case ... ... ( ... hmmm ... pity Tu didn't use \(\displaystyle u_1, u_2, \ ... \ ... \ , u_n\) as the basis for \(\displaystyle V\) ... ... )My suspicions about the basis being not general ... but indeed the standard basis ... ... came about on reading the following note on page 11 concerning notation ...

View attachment 8793Hope someone can clarify the above ...

Help will be appreciated ...