An unbounded symmetric operator on an An unbounded symmetric operator on an inner product space.

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Suppose that H is the subspace of l2 consisting of all sequences (zn) with zn = 0 for all sufficiently larg n.  H is not complete ( Since (an) where an = (1, [1/2], ¼,[1/n], 0, 0, ¼)n Î N is a Cauchy divergent sequence in H).
Let T denote the linear mapping (zn) ® (nzn) on H.  T is symmetric, for < T((zn)) , (hn) > = ån = 1¥ nzn[`(hn)] = < (zn) , T((hn)) > . T is unbounded since if (xn) is the orthonormal basis for l2, for each n, xn Î H, ||xn || = 1 and ||Txn|| = n.


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On 22 Feb 2001, 00:21.