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Let H be a separable Hilbert space with the standard orthonormal basis (xn). Define T on H by Txn = nx1 and extend T to the dense linear subspace D(T) of finite linear combinations of basis elements xn ( we denote the extension of T by the same T ). Then T is a densely defined unbounded operator on H ( since limn ® 0 [(||Txn ||)/(||xn ||)] = limn ® 0 n = ¥). Moreover T is not closable, for limn ® 0 [(xn)/n] = 0 but limn ® 0T([(xn)/n]) = x1.