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Homework Help: Isometrically isomorphic normed spaces

  1. Jun 21, 2008 #1
    Let X and Y be normed spaces. If X and Y are isometrically isomorphic, then their duals X' and Y' are also isometrically isomorphic.

    I have no idea what to do with this, please help. :(
     
  2. jcsd
  3. Jun 21, 2008 #2

    morphism

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    This is just a matter of writing down what you have, and understanding the definitions. Say you have an isometric isomorphism F:X->Y. We want to get an isometric isomorphism G:X'->Y'. The question you should be asking yourself is: If we start with an element f in X', how can we use this to get an element G(f) in Y'?
     
  4. Jun 21, 2008 #3

    matt grime

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    Duality if contravariant. Surely a morphism F:X->Y more naturally (no pun) gives a map G:Y'->X'.
     
  5. Jun 21, 2008 #4

    morphism

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    Right. I had initially written my G as going from Y' to X', but decided to stick with X'->Y' in case the OP doesn't immediately see why we've switched directions. But regardless, the moment he/she sets up the obvious map, he/she will probably notice that it's easier to go from Y' to X'.
     
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