An unbounded symmetric operator on an inner product space.

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Suppose that H is the subspace of l^{2} consisting of all sequences (ζ_{n}) with ζ_{n} = 0 for all sufficiently larg n. H is not complete ( Since (a_{n}) where a_{n} = (1, \frac{1}{2}, ⋯, \frac{1}{n}, 0, 0, ⋯)_{n∈N} is a Cauchy divergent sequence in H). Let T denote the linear mapping (ζ_{n}) ↦ (nζ_{n}) on H. T is symmetric, for < T ((ζ_{n})), (η_{n}) >= ∑_{n=1}^{∞}nζ_{n}\overline{η_{n}} =< (ζ_{n}), T ((η_{n})) > . T is unbounded since if (ξ_{n}) is the orthonormal basis for l^{2}, for each n, ξ_{n} ∈ H, ∥ ξ_{n} ∥= 1 and ∥ T ξ_{n} ∥= n.