COUNTEREXAMPLES IN BANACH AND HILBERT SPACES

COUNTEREXAMPLES IN BANACH AND HILBERT SPACES

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