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.

 

[blue_raver22]

I would like to address this problem by first defining some relevant terms. A smooth manifold is a mathematical concept that describes a space that is locally similar to Euclidean space. It is an important tool in many areas of mathematics and physics, including differential geometry and general relativity. In this context, C^\inf(M) refers to the space of smooth functions on the manifold M, and (T_p(M))^star is the dual space of the tangent space at point p on the manifold.

Now, let's consider the given map \phi: F_p \rightarrow (T_p(M))^star, where f \in F_p is mapped to its differential df_p. This map is well-defined because f vanishes at p, so df_p is a linear map from the tangent space to the real numbers. We can also see that \phi is a surjective map, since any linear map from the tangent space to the real numbers can be written as df_p for some smooth function f.

Next, we want to show that F_p/F^2_p is isomorphic to (T_p(M))^star. This means that there exists a bijective map between these two spaces that preserves their algebraic structure. To prove this, we can define a map \psi: F_p/F^2_p \rightarrow (T_p(M))^star by \psi([f]) = df_p, where [f] is the equivalence class of f in the quotient space. This map is well-defined because if [f] = [g], then f-g \in F^2_p, which means df_p = dg_p. It is also easy to see that \psi is a bijection, since it has an inverse given by \phi.

Moreover, we can show that \psi preserves the algebraic structure by showing that \psi([f+g]) = d(f+g)_p = df_p + dg_p = \psi([f]) + \psi([g]) and \psi([af]) = d(af)_p = adf_p = a\psi([f]), where a is a scalar. This proves that \psi is a linear map, and thus, F_p/F^2_p is isomorphic to (T_p(M))^star.

In conclusion, we have shown that the quotient space F_p/F^2_p is isomorphic to the dual space (T_p(M))^star by defining a map \psi that preserves the