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The purpose of this paper is to present an example of a C1 (in the Fr\'echetsense) discrete dynamical system in a infinite-dimensional separable Hilbertspace for which the origin is an exponentially asymptotically stable fixedpoint, but such that its derivative at the origin has spectral radius largerthan unity, and this means that the origin is unstable in the sense of Lyapunovfor the linearized system. The possible existence or not of an example of thiskind has been an open question until now, to our knowledge. The construction isbased on a classical example in Operator Theory due to Kakutani.

An example on Lyapunov stability and linearization