COUNTEREXAMPLES IN BANACH AND HILBERT SPACES |
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PROBLEMS | SOLUTIONS |
BH1. A Banach space with a non-complemented closed subspace. |
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BH2. A complete metrizable linear space whose metric cannot be obtained from a norm. |
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BH3. Two non-isometrically isomorphic spaces with the same duals. So that a such dual space could not be a W*-algebra under any multiplication and involution. |
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BH4. A Banach space X such that all its closed subspaces are complemented. |
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BH5. A Banach space which isn't metrizable in weak topology. |
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BH6. A Banach space which is not an inner product space. |
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BH7. An incomplete inner product space. |
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BH8. Two closed densely defined operators T and S on a Hilbert space such that T + S isn't closable. |
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BH9. A Hilbert space whose Hamel dimension and Hilbert dimension are different. |
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BH10. A nonclosable unbounded operator on a Hilbert space. |
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BH11. On a separable infinite dimensional Banach space X there exists another norm under which A isn't separable. |
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