Abstract: then F is called the Frtchet derivative. In finite-dimensional vector spaces it is called the directional derivative. The idea of the FrCchet derivative can be extended to both topological and nonnormed pseudotopological linear vector spaces, where it is called a functional derivative. The details of this extension, which is based on the concept of a remainder, can be found in [3]. The classical idea of the gradient can bc defined as follows. Let the primary topological space be a Hilbert space and let g, be a countable complete orthonormal set of vectors. Then the function,