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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 $$e_1, \ ... \ ... \ e_n$$ be a basis for $$V$$. ... ... "
Now my question is as follows:Is $$e_1, \ ... \ ... \ e_n$$ a completely general basis ... hence making Proposition 3.1 completely general (in terms of basis anyway) ... or given the notation is $$e_1, \ ... \ ... \ e_n$$ the standard basis where $$e_1 = ( 1, 0, 0, \ ... \ ... \ , 0 )^T , e_2 = ( 0,1, 0, \ ... \ ... \ , 0 )^T , \ ... \ ... \ , \ e_n = ( 0, 0, 0, \ ... \ ... \ , 1 )^T$$ ...I must say I can find no instance in the proof of Proposition 3.1 where the basis $$e_1, \ ... \ ... \ e_n$$ 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 $$u_1, u_2, \ ... \ ... \ , u_n$$ as the basis for $$V$$ ... ... )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 $$e_1, \ ... \ ... \ e_n$$ be a basis for $$V$$. ... ... "
Now my question is as follows:Is $$e_1, \ ... \ ... \ e_n$$ a completely general basis ... hence making Proposition 3.1 completely general (in terms of basis anyway) ... or given the notation is $$e_1, \ ... \ ... \ e_n$$ the standard basis where $$e_1 = ( 1, 0, 0, \ ... \ ... \ , 0 )^T , e_2 = ( 0,1, 0, \ ... \ ... \ , 0 )^T , \ ... \ ... \ , \ e_n = ( 0, 0, 0, \ ... \ ... \ , 1 )^T$$ ...I must say I can find no instance in the proof of Proposition 3.1 where the basis $$e_1, \ ... \ ... \ e_n$$ 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 $$u_1, u_2, \ ... \ ... \ , u_n$$ as the basis for $$V$$ ... ... )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 ...