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 = l^{2}. Extend (e_{n}) to a Hamel basis β for l^{2}. Choose f ∈ β distinct to the e_{n} and define then the linear mapping T : H ⟶ H by

T (g) = \{ \begin{matrix}1 & g = f \\ 0 & g ∈ β \ \{f\} \end{matrix}

Then T (e_{n}) = 0 and T is unbounded ( otherwise, 1 = T (f) = ∑_{n=1}^{∞} < f, e_{n} > T e_{n} = 0).