It shows that V is the dual space of V*?

[Mr.M]

Not my homework
It's in the textbook - lectures of diff geo by s.s.chern
Just put them down in a clearer way
Could anybody explain the questions in the pic ?
http://x7d.xanga.com/be6d931344030137094065/z100635368.jpg"

 

[matt grime]

This is only true if V is finite dimensional. Hint just write down bases.

 

[Mr.M]

actually i don't understand the last step
Phi is taken out of the sum because it is linear in the 2nd variable ?
And "V is the dual space of V*"looks very confusing
The dual space of V is the set linear functions on V
Now the dual space of V* is the set linear functions on V*?
how can i see that ?

 

[Mr.M]

i just got another idea
since <v,v*> just depends on v*
so we just treat it as a function of v* and then we have the last step ?

 

[matt grime]

If V is a finite dimensional vector space, then V the only invariant of V is its dimension. ALL vector spaces of a given dimension (over the same field) are isomorphic. Since V is clearly isomorphic to V* by picking a basis and corresponding dual basis, this shows that (V*)* must be isomorphic to V since they have the same basis. (This fails in infinite dimensions.)

What I presume you're looking at is a 'nice' bijection that sends v in V to the function on V* that sends f to f(v). Notice how an element of v can be sent to a function, call it e_v, and think of it as evaluation at v. This is a linear functional on the space of linear functionals:

e_v(f)=f(v).

This defines a map from V to the space of linear functionals on V*

v--->e_v

it is straight foward to show this is an isomorphism.

 

[mathwonk]

if f(x) is a number, then f sends x to a number, and x sends f to a number. so f is a function of x, and x is a function of f. so points of the domain are functions on functions.