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  1. Prove that a closed subspace of a Banach space is also a Banach space.
  2. Show that the linear space of all polynomials in one variable is not a Banach space in any norm.
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    @DavidMitra: it's not just you :)2012-04-10

1 Answers 1

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Hints:

  1. Prove that a closed subspace of a complete metric space is complete.

  2. The subspace of polynomials of degree $\leq n$ is closed in any norm because it is finite-dimensional. Hence the space of all polynomials can be written as countable union of closed nowhere dense sets. If there were a complete norm this would contradict the Baire category theorem.

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    Aye! *closes tab* (for now)2012-07-29