Leray–Schauder degree

In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres ${\displaystyle (S^{n},*)\to (S^{n},*)}$ or equivalently to boundary-sphere-preserving continuous maps between balls ${\displaystyle (B^{n},S^{n-1})\to (B^{n},S^{n-1})}$ to boundary-sphere-preserving maps between balls in a Banach space ${\displaystyle f:(B(V),S(V))\to (B(V),S(V))}$, assuming that the map is of the form ${\displaystyle f=id-C}$ where ${\displaystyle id}$ is the identity map and ${\displaystyle C}$ is some compact map (i.e. mapping bounded sets to sets whose closure is compact).^[1]

The degree was invented by Jean Leray and Juliusz Schauder to prove existence results for partial differential equations.^[2][3]