How Is F_p/F^2_p Isomorphic to (T_p(M))^* in Smooth Manifolds?

[seydunas]

Hi,

I want to ask a problem from Lee ' s book Introduction to Smooth Manifolds: Let F_p denote the subspace of C^\inf(M) consisting of smooth functions that vanish at p and let F^2_p be the subspace of F_p spanned by functions fg for some f,g \in F_p. Define a map \phi: F_p-----> (T_p(M))^star by f-----> df_p Show that F_p/F^2_p is isomorphic to (T_p(M))^star

Thanks...

 

[quasar987]

By the first isomorphism theorem in algebra, you only need to show that ker(phi)=F²_p.

 

[seydunas]

I can see why ker(phi)=F²_p. But the first isomorphism theorem of algebra is valid for groups, right? Ohh yes, every vector space has a group structure under addition. So, we are done.

 

[seydunas]

Sorry, our map is linear. So this is true for vector spaces. Now i understood everything.