In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (S^n, *) \to (S^n , *)} or equivalently to a 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]

References