mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Leray–Schauder degree is an extension of the degree of a

base point In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...

preserving

continuous map In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

between

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

(S^n, *) \to (S^n , *)

or equivalently to boundary-sphere-preserving continuous maps between balls

(B^n, S^) \to (B^n, S^)

to boundary-sphere-preserving maps between balls in a

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

f: (B(V), S(V)) \to (B(V), S(V))

, assuming that the map is of the form

f = id - C

where

id

is the

identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

and

C

is some compact map (i.e. mapping bounded sets to sets whose closure is

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

). The degree was invented by

Jean Leray Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...

and

Juliusz Schauder Juliusz Paweł Schauder (; 21 September 1899 – September 1943) was a Polish mathematician known for his work in functional analysis, partial differential equations and mathematical physics. Life and career Born on 21 September 1899 in Lwów ...

to prove existence results for partial differential equations.Mawhin, J. (2018). A tribute to Juliusz Schauder. ''

Antiquitates Mathematicae The Polish Mathematical Society () is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). History The society wa ...

'', ''12''.

References

Topology {{DEFAULTSORT:Leray-Schauder degree