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We know they’re isomorphic, but how are they isomorphic?

14 October, 2018

The given information

  1. L \cong L \oplus L, in a very nice way
  2. m \cong m \oplus m, in a very nice way
  3. L\cong m \oplus Q_m, in a nice way
  4. m \cong L \oplus Q_L, in a faintly dodgy way

Wizardry from Warsaw

m \cong L\oplus Q_L \cong (L\oplus L) \oplus Q_L \cong L \oplus (L\oplus Q_L) \cong L \oplus m

L \cong m\oplus Q_m \cong (m\oplus m) \oplus Q_m \cong m\oplus (m\oplus Q_m) \cong m\oplus L

Conclusion: m \cong L\oplus m \cong m\oplus L \cong L.

A nagging question

Just how nice or dodgy is our final isomorphism m\cong L?

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