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Lee 6.7

  1. Nov 19, 2011 #1
    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...
     
  2. jcsd
  3. Nov 19, 2011 #2

    quasar987

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    By the first isomorphism theorem in algebra, you only need to show that ker(phi)=F²_p.
     
  4. Nov 20, 2011 #3
    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.
     
  5. Nov 20, 2011 #4
    Sorry, our map is linear. So this is true for vector spaces. Now i understood everything.
     
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