Operators of arbitrary large norms that are bounded by 1 on a given basis of a separable infinite dimensional Hilbert space H.

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Let (ξ_{n}) be an orthonormal basis for H. For k ∈ N, define T_{k} on H by T_{k}η =< η, ξ_{1} + ξ_{2} + ⋯ + ξ_{k} > ξ_{1}. Then

T_{k}ξ_{n} = \{ \begin{matrix}ξ_{1} & n ≤ k \\ 0 & n > k \end{matrix}

Hence ∥ T_{k}ξ_{n} ∥≤ 1(n ∈ N). On the other hand T_{k}^{∗}η =< η, ξ_{1} > (ξ_{1} + ⋯ + ξ_{k}) (η ∈ H); therefore ∥ T_{k} ∥=∥ T_{k}^{∗} ∥≥∥ T_{k}^{∗}ξ_{1} ∥=∥ ξ_{1} + ⋯ + ξ_{k} ∥= \sqrt{k}.