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

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Discussion Overview

The discussion revolves around a problem from Lee's "Introduction to Smooth Manifolds" concerning the isomorphism between the quotient space F_p/F^2_p and the dual space (T_p(M))^*. Participants explore the properties of the map defined from F_p to (T_p(M))^* and the application of the first isomorphism theorem in this context.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents the problem and defines the relevant spaces and map.
  • Another participant suggests that demonstrating ker(phi) = F²_p is sufficient to establish the isomorphism using the first isomorphism theorem.
  • A later reply questions the applicability of the first isomorphism theorem, noting its typical context in group theory, but acknowledges that vector spaces also have a group structure under addition.
  • Subsequently, a participant clarifies that the linearity of the map allows the use of the first isomorphism theorem for vector spaces, indicating a resolution of their earlier confusion.

Areas of Agreement / Disagreement

Participants generally agree on the approach to the problem and the application of the first isomorphism theorem, though there is a moment of uncertainty regarding its context in vector spaces versus groups.

Contextual Notes

The discussion does not address any potential limitations or assumptions explicitly, but the reliance on the first isomorphism theorem suggests a need for clarity on the conditions under which it applies to the specific spaces involved.

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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