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Why are $T^k(V)$ and $V^* otimes… otimes V^*$ just isomorphic?

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In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* otimes... otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $epsilon^1,...,epsilon^n$ is its dual basis than $epsilon^{i_1}otimes...otimesepsilon^{i_n}$ is a basis of both $T^k(V)$ and $V^* otimes... otimes V^*$ no? So why just an isomorphism?

tensor-products tensors dual-spaces vector-space-isomorphism

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asked Mar 13 at 16:15

roi_saumonroi_saumon

61838

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  First, see the definition of $T^k(V)$ on pages 172-173. Then go to Proposition 8.4(a) and find the proof for $k=2$, where you can see the actual isomorphism.
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  – Zeekless
  Mar 13 at 16:35
  

  
  
  
  


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  The version of my book that you linked to is a pirated early draft of the first edition, which somebody posted illegally on the internet. It's full of mistakes and comes with no guarantees.
  $endgroup$
  – Jack Lee
  Mar 17 at 17:37
  

  
  
  
  

add a comment | 

0

$begingroup$

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* otimes... otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $epsilon^1,...,epsilon^n$ is its dual basis than $epsilon^{i_1}otimes...otimesepsilon^{i_n}$ is a basis of both $T^k(V)$ and $V^* otimes... otimes V^*$ no? So why just an isomorphism?

tensor-products tensors dual-spaces vector-space-isomorphism

share|cite|improve this question

asked Mar 13 at 16:15

roi_saumonroi_saumon

61838

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  First, see the definition of $T^k(V)$ on pages 172-173. Then go to Proposition 8.4(a) and find the proof for $k=2$, where you can see the actual isomorphism.
  $endgroup$
  – Zeekless
  Mar 13 at 16:35
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The version of my book that you linked to is a pirated early draft of the first edition, which somebody posted illegally on the internet. It's full of mistakes and comes with no guarantees.
  $endgroup$
  – Jack Lee
  Mar 17 at 17:37
  

  
  
  
  

add a comment | 

$begingroup$

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* otimes... otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $epsilon^1,...,epsilon^n$ is its dual basis than $epsilon^{i_1}otimes...otimesepsilon^{i_n}$ is a basis of both $T^k(V)$ and $V^* otimes... otimes V^*$ no? So why just an isomorphism?

tensor-products tensors dual-spaces vector-space-isomorphism

share|cite|improve this question

asked Mar 13 at 16:15

roi_saumonroi_saumon

61838

$endgroup$

share|cite|improve this question

asked Mar 13 at 16:15

roi_saumonroi_saumon

61838

share|cite|improve this question

asked Mar 13 at 16:15

roi_saumonroi_saumon

61838

asked Mar 13 at 16:15

roi_saumonroi_saumon

61838

asked Mar 13 at 16:15

roi_saumonroi_saumon

61838