A nonclosable unbounded operator on a Hilbert space.

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Let H be a separable Hilbert space with the standard orthonormal basis (ξ_{n}). Define T on H by T ξ_{n} = nξ_{1} and extend T to the dense linear subspace D(T) of finite linear combinations of basis elements ξ_{n} ( we denote the extension of T by the same T ). Then T is a densely defined unbounded operator on H ( since lim_{n→0} \frac{∥ T ξ_{n} ∥}{∥ ξ_{n} ∥} = lim_{n→0}n = ∞). Moreover T is not closable,

for lim_{n→0} \frac{ξ_{n}}{n} = 0 but lim_{n→0}T (\frac{ξ_{n}}{n}) = ξ_{1}.