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I'm trying to understand the natural transformation from V to V**, and the book has the theory but I think I'm needing an example.

Lets say V=R^2 a vector space over K=R.

B={(1,1),(1,-1)} a basis of V

B={x/2 + y/2, x/2 - y/2} a basis of V*

v = (3,2) a vector of V

I want to get a vector of V** (a funtional of V*), it is supposed to be

Lf_v = f(v)

with f in V*

But who is f? a generic funtional? lets say

f=ax+by

then

f(v) = 3a + 2b ?

then Lv = 3a + 2b?

And I cant also see how to construct a basis for V**

Please, any help will be appreciated. Thanks!