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## Summary:

- Mathematically, for finite dimensional vector spaces, there is a natural isomorphism between a vector space V and its Double Dual V**. Does this men that V** is always the original V ? Or simply that V** is a vector space with the same properties or structure as V ?

## Main Question or Discussion Point

Hi

I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**.

What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical meaning it IS V) ? In other words have you come right back to the original vector space V ?

In the book General Relativity (by Robert M Wald) he writes the following (page 20) - "Thus taking the double dual gives nothing new. We can naturally identify V** with the original vector space V. This identification will be assumed in the discussion below".

Is this always the case or is he just assuming this is the case in his particular context ?

Thank you !

I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**.

What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical meaning it IS V) ? In other words have you come right back to the original vector space V ?

In the book General Relativity (by Robert M Wald) he writes the following (page 20) - "Thus taking the double dual gives nothing new. We can naturally identify V** with the original vector space V. This identification will be assumed in the discussion below".

Is this always the case or is he just assuming this is the case in his particular context ?

Thank you !