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

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SUMMARY

The discussion centers on the isomorphism between the quotient space F_p/F^2_p and the dual space (T_p(M))^* in the context of smooth manifolds, as outlined in Lee's "Introduction to Smooth Manifolds." The map φ: F_p → (T_p(M))^* defined by φ(f) = df_p is crucial for this proof. By applying the first isomorphism theorem, it is established that the kernel of φ is indeed F^2_p, confirming the isomorphism. The linearity of the map φ further supports the validity of this conclusion within vector spaces.

PREREQUISITES
  • Understanding of smooth manifolds and the notation used in differential geometry.
  • Familiarity with the concepts of dual spaces and linear maps.
  • Knowledge of the first isomorphism theorem in algebra.
  • Basic grasp of the structure of function spaces, particularly C^\inf(M).
NEXT STEPS
  • Study the properties of dual spaces in the context of smooth manifolds.
  • Explore the first isomorphism theorem and its applications in vector spaces.
  • Investigate the structure of the space C^\inf(M) and its subspaces.
  • Learn about the implications of linear maps in differential geometry.
USEFUL FOR

This discussion is beneficial for graduate students in mathematics, particularly those specializing in differential geometry, as well as researchers exploring the properties of smooth manifolds and their function spaces.

seydunas
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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...
 
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By the first isomorphism theorem in algebra, you only need to show that ker(phi)=F²_p.
 
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.
 
Sorry, our map is linear. So this is true for vector spaces. Now i understood everything.
 

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