- Prove that a closed subspace of a Banach space is also a Banach space.
- Show that the linear space of all polynomials in one variable is not a Banach space in any norm.
Banach space in functional analysis
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functional-analysis
banach-spaces
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4Welcome to math.SE! What have you tried so far? For example, for the first problem, you have to show that a closed subspace is complete with respect to the norm inherited from the norm on the initial Banach. But in fact it doesn't have anything to do with vector space (it's true that a closed part of a complete metric space is complete). For the second exercise, you can use Baire's categories theorem. – 2012-04-10
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7Is it just me or does one of these problems seem substantially more difficult than the other? – 2012-04-10
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2@DavidMitra: it's not just you :) – 2012-04-10