The study of linear functionals, as an important special case of linear
transformations, is one of the key topics in linear algebra and plays a
significant role in analysis. In this paper we generalize the crucial
results from the classical theory and study main properties of linear
functionals on hypervector spaces. In this way, we obtain the dual basis of
a given basis for a finite-dimensional hypervector space. Moreover, we
investigate the relation between linear functionals and subhyperspaces and
conclude the dimension of the vector space of all linear functionals over a
hypervector space, the dimension of sum of two subhyperspaces and the
dimension of the annihilator of a subhyperspace, under special conditions.
Also, we show that every superhyperspace is the kernel of a linear
functional. Finally, we check out whether every basis for the vector space
of all linear functionals over a hypervector space V is the dual of some
basis for V.