A reduction theorem for the topological degree for mappings of class $(S+)$

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class $(S_+)$ from a real separable reflexive Banach space $X$ into the dual space $X^*$ can be reduced into the calculation of degree of mapping from a closed subspace $V\subset X$ into $V^*.$ Since the Leray-Schauder mappings are acting from $X$ to $X$ and we are dealing with mappings from $X$ to $X^*,$ the standard `invariant subspace' condition must be replaced by a less obvious one.