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