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Consider $E=C([0,1])$ as a vector space. Is there a norm $N$ on $E$ such that

  • $(E,N)$ is Banach,
  • $N$ is not equivalent to the usual norm $N_\infty$?
Tomasz Kania
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Phil-W
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    Yes, see here for something similar. But I'd be very very surprised if someone could explicitly construct such a norm. – Daniel Fischer Nov 15 '16 at 22:54
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    @DanielFischer: In a sense it can't be done explicitly, in that it is consistent with ZF+DC that no other such norm exists. It's consistent with ZF+DC that all linear maps between Banach spaces are continuous, and if the identity map from $(E, N_\infty)$ to $(E,N)$ is continuous then by the open mapping theorem it's a homeomorphism, meaning $N, N_\infty$ are equivalent. – Nate Eldredge Nov 15 '16 at 23:00

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