In this paper, we propose a new energy efficient neural network with the universal approximation property over space of Lebesgue integrable functions. This network, called additive neural network, is very suitable for mobile computing. The neural structure is based on a novel vector product definition, called ef-operator, that permits a multiplier-free implementation. In ef-operation, the "product" of two real numbers is defined as the sum of their absolute values, with the sign determined by the sign of the product of the numbers. This "product" is used to construct a vector product in n-dimensional Euclidean space. The vector product induces the lasso norm. The proposed additive neural network successfully solves the XOR problem. The experiments on MNIST dataset show that the classification performances of the proposed additive neural networks are very similar to the corresponding multi-layer perceptron.