Bijection between tangent spaces
We have defined the tangent space at a point p of a manifold as the set of all derivatives.
Now i want to show, that there exist a bijection between all derivatives and the "Tangentvectors" defined by (equivalent classes of) paths.
We call two paths equivalent, iff they define the same derivation at a point p. Now i want to show that there is a bijection between the set of all equivalent classes and the Tangentspace (defined as the set of derivatives).
My Approach was the following.
Every path c:I->M into the manifold M defines a derivation by
so i try to show that with is a bijection.
Of course is well defined and injective, but i couldn't show that
If D is some arbitrary derivative at the point p. How can i find a path c, s.t. ?