By conjugate linear transformation, I mean under scalar multiplication instead of C ( a f ) = a C ( f ), I would have C ( a f ) = a ¯ C ( f ), where a is a constant complex number, and C is the transformation.

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By conjugate linear transformation, I mean under scalar multiplication instead of   C  (  a  f  )  =  a  C  (  f  ), I would have   C  (  a  f  )  =      a    ¯    C  (  f  ), where a is   a constant complex number, and   C is the transformation.

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Given a linear-conjugate function   C  :  V  ⟶  V,   V a complex vector space, we define the conjugate vector space      V    ¯  , by being the same set   V, with an altered multiplication by scalar ∗, defined by  λ  ∗  x  :=      λ    ¯    ⋅  xwhere   ⋅ is   V&#039;s usual multiplication. See the notations,   V  =  (  V  ,  +  ,  ⋅  ), and       V    ¯    =  (  V  ,  +  ,  ∗  ). Reforce, the set of vectors is the same. It should be checked that       V    ¯   is indeed a vector space, although it&#039;s easy to see. Some facts are:i) a set   B is a basis for   V iff it is a basis for       V    ¯  ;ii) from i) follows that   dim  ⁡  V  =  dim  ⁡      V    ¯  ;iii)   W is a subspace of   V iff it is a subspace of       V    ¯  ;iv) Given a basis   B,   x  ∈  V has coordinates   (      x    1    ,  ⋯  ,      x    n        )      B   iff considering   x  ∈      V    ¯  , we have   x  =  (            x      1        ¯    ,  ⋯  ,            x      n        ¯        )      B  v)   C  :  V  →  V is linear conjugate iff   C  :      V    ¯    →  V is linear iff   C  :  V  →      V    ¯   is linear;vi)             V      ¯        ¯    =  V

By conjugate linear transformation, I mean under scalar multiplication instead of C ( a

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