Boundedness and Invertibility

  • Paul R. Halmos
Part of the Graduate Texts in Mathematics book series (GTM, volume 19)


Let {e1 e2, e3,…} be an orthonormal basis for a Hubert space H, and find a Hamel basis for H that contains each e n . Let f0 be an arbitrary but fixed element of that Hamel basis distinct from each e n (see Solution 7). A unique linear transformation A is defined on H by the requirement that Af 0 = f0 and Af = 0 for all other elements of the selected basis: in particular, Ae n = 0 for all n. If A were bounded, then its vanishing on each e n would imply that A = 0. This solves parts (a) and (b) of the problem.


Hilbert Space Linear Transformation Borel Subset Cardinal Number Polar Decomposition 
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© Springer-Verlag New York Inc. 1982

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  • Paul R. Halmos

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