Double Dual Example: V=R^2 to V** Transformation

[Damidami]

Hi,
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? let's say
f=ax+by
then
f(v) = 3a + 2b ?

then Lv = 3a + 2b?

And I can't also see how to construct a basis for V**

Please, any help will be appreciated. Thanks!

 

[fresh_42]

Lets say we have a basis ##\{\,v_k\,\}## of ##V##. Then we get a basis for ##V^*## by ##\{\,f_k\, : \, v_m \longmapsto \langle v_m,v_k \rangle\,\}##. In coordinates the two are indistinguishable, their usage changed. Repeating this process for ##V^*## as basic vector space leads you automatically back to ##V##.

 

[mathman]

Do you have a question?

 

[lavinia]

If ##v## is a vector in ##V## then the rule ##l→l(v)## defines an element of ##V^{**}##. That is: the evaluation map at a fixed vector defines a linear map of the dual vector space into the base field. This mapping of ##V## into ##V^{**}## is natural because it is defined without a basis for ##V##.

##V## is also isomorphic to ##V^{*}## but there is no natural isomorphism.