A theoretical analysis, using the properties of linear vector spaces, is applied to the design of a nonlinear equaliser. The design minimises the probability of error in the detection of a serial digital signal, subject to the accurate equalisation of the time-invariant channel, when the signal is received in the presence of additive white Gaussian noise with a high signal/noise ratio. The equaliser contains a conventional combination of linear feedforward and nonlinear feedback transversal filters. Certain commonly used assumptions are made concerning the transmitted signal and the channel, but it is not assumed that the channel can necessarily be equalised by a linear feedforward transversal filter of limited length. A simple example shows the advantage in tolerance to additive Gaussian noise, which is achieved by the nonlinear equaliser over the corresponding linear equaliser, in a particular case where accurate equalisation of the channel is achieved by the linear equaliser.