Star operator in the simplest formLet E together with g be a inner product space(over field R) , dim E = n &lt; ∞ and { e 1 , ⋯ , e n } is orthonormal basis of E that { e 1 , ⋯ , e n } is its dual basis(for E∗). Now we define ω := e 1 ∧ ⋯ ∧ e n as an element of volume of E.I prove that g ♭ : E → E ∗ with rule ( g ♭ ( u ) ) ( v ) = g ( u , v ), is an isomorphism ∀ u , v ∈ EConvention: u ~ := g ♭ ( u )I wish to prove that for any p-form θ ∈ Λ p ( E ), There exist a unique element η ∈ Λ ( n − p ) ( E ) such that η ( u 1 , ⋯ , u n − p ) ω = θ ∧ u ~ 1 ⋯ ∧ u ~ n − p ∀ u 1 , ⋯ , u n − p ∈ EHow can I do this?Of course I guess that should be defined an inner product g Λ k ( E ) on Λ k ( E ) for any 0&lt;k&lt;n and then use g Λ k ( E ) ♭ . what is your method ? and How?

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Star operator in the simplest formLet E together with g be a inner product space(over field R) ,   dim  E  =  n  &amp;lt;  ∞ and   {      e    1    ,  ⋯  ,      e    n    } is orthonormal basis of E that   {      e    1    ,  ⋯  ,      e    n    } is its dual basis(for E∗). Now we define   ω  :=      e    1    ∧  ⋯  ∧      e    n   as an element of volume of E.I prove that       g          ♭        :  E  →      E    ∗   with rule   (      g          ♭        (  u  )  )  (  v  )  =  g  (  u  ,  v  ), is an isomorphism   ∀  u  ,  v  ∈  EConvention:             u      ~        :=      g          ♭        (  u  )I wish to prove that for any p-form   θ  ∈      Λ    p    (  E  ), There exist a unique element   η  ∈      Λ          (      n      −      p      )        (  E  ) such that  η  (      u    1    ,  ⋯  ,      u          n      −      p        )  ω  =  θ  ∧                    u        ~              1    ⋯  ∧                    u        ~                    n      −      p          ∀      u    1    ,  ⋯  ,      u          n      −      p        ∈  EHow can I do this?Of course I guess that should be defined an inner product       g                  Λ        k            (      E      )       on       Λ    k    (  E  ) for any 0&amp;lt;k&amp;lt;n and then use       g                  Λ        k            (      E      )              ♭      . what is your method ? and How?

illpnthr21vw · Accepted Answer

Given a p-form   θ  ∈      ⋀    p        E    ∗  , we can define an alternating multilinear map   h  :      E          n      −      p        →      ⋀    n    E by  h  (      u    1    ,  …  ,      u          n      −      p        )  =  θ  ∧                    u        ~              1    ∧  …  ∧                    u        ~                    n      −      p        .Let   b  :      R    →      ⋀    n    E be the linear map  b  (  t  )  =  t  ω  .Because       ⋀    n    E is one-dimensional and   ω is nonzero, this map is an isomorphism. You can think of it as the coordinate system for       ⋀    n    E given by the basis   {  ω  }. Observe that, for any   α  ∈      ⋀    n    E, the scalar       b          −      1        (  α  ) is the unique scalar with the property that       b          −      1        (  α  )    ω  =  αComposing h with       b          −      1       gives a map   H  :      E          n      −      p        →      R  . Looking back at our discussions of h and       b          −      1      , you can see that   H  (      u    1    ,  …  ,      u          n      −      p        ) is the unique scalar with the property that  H  (      u    1    ,  …  ,      u          n      −      p        )    ω  =  θ  ∧                    u        ~              1    ∧  …  ∧                    u        ~                    n      −      p        .Since h is multilinear and alternating, H is too. The map H can therefore be expressed as an   (  n  −  p  ) -form   η, using the standard correspondence between elements of       E          n      −      p        →      R   and alternating multilinear maps       E          n      −      p        →      R

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