In nonlinear functional analysis, the Krasnoselskii genus generalizes the notion of dimension for vector spaces. The Krasnoselskii genus of a linear space formula_1 is the smallest natural number formula_2 for which there exists a continuous odd function of the form formula_3. The genus was introduced by Mark Aleksandrovich Krasnoselskii in 1964, and an equivalent definition was provided by Charles Coffman in 1969.
Krasnoselskii Genus.
We follow the definition given by Coffman.
Let
For formula_9 define the set
formula_10
Then the Krasnoselskii genus of formula_1 is defined as
formula_12
In other words, if formula_13 then there exists a continuous odd function formula_14 such that formula_15. Moreover formula_2 is the minimal possible dimension, i.e. there exists no such function formula_17 with formula_18.
Properties.
Combining these statements, it follows immediately that if there exists an odd homeomorphism between formula_1 and formula_32 then formula_13.