An unbounded operator on a Hilbert

An unbounded operator on a Hilbert An unbounded operator on a Hilbert space H annihilating an orthonormal basis of H.

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Let (e_n) be the standard orthonormal basis for the Hilbert space H = l². Extend (e_n) to a Hamel basis b for l². Choose f Î b distinct to the e_n and define then the linear mapping T: H ® H by

T(g) = ì
í
î

1
g = f

0
g Î b\{f}

Then T(e_n) = 0 and T is unbounded ( otherwise, 1 = T(f) = å_{n = 1}^¥ < f,e_n > Te_n = 0).

File translated from T_EX by T_TH, version 2.70.
On 22 Feb 2001, 00:22.