In nonlinear functional analysis, the Krasnoselskii genus generalizes the notion of dimension for vector spaces. The Krasnoselskii genus of a linear space A {\displaystyle A} is the smallest natural number n {\displaystyle n} for which there exists a continuous odd function of the form f : A → R n ∖ 0 {\displaystyle f:A\to \mathbb {R} ^{n}\setminus {0}} . 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
E {\displaystyle E} be a Banach space,
A = { A ⊂ E : A closed , ; A = − A } {\displaystyle {\mathcal {A}}=\{A\subset E:A{\text{ closed}},;A=-A\}} be the collection of symmetric closed subsets of E {\displaystyle E} ,
C ( A , R n ) {\displaystyle C(A,\mathbb {R} ^{n})} the space of continuous functions A → R n {\displaystyle A\to \mathbb {R} ^{n}} . For A ∈ A {\displaystyle A\in {\mathcal {A}}} define the set
K A = { n ∈ N : ∃ f ∈ C ( A , R n ∖ 0 ) , ; f ( − x ) = − f ( x ) } {\displaystyle K_{A}=\{n\in \mathbb {N} :\exists f\in C(A,\mathbb {R} ^{n}\setminus {0}),;f(-x)=-f(x)\}}
Then the Krasnoselskii genus of A {\displaystyle A} is defined as
γ ( A ) = { inf K A if K A ≠ ∅ , ∞ if K A = ∅ , 0 if A = ∅ . {\displaystyle \gamma (A)={\begin{cases}\inf K_{A}&{\text{if }}K_{A}\neq \emptyset ,\\\infty &{\text{if }}K_{A}=\emptyset ,\\0&{\text{if }}A=\emptyset .\end{cases}}}
In other words, if γ ( A ) = n {\displaystyle \gamma (A)=n} then there exists a continuous odd function φ : A → R n {\displaystyle \varphi :A\to \mathbb {R} ^{n}} such that 0 ∉ φ ( A ) {\displaystyle 0\notin \varphi (A)} . Moreover n {\displaystyle n} is the minimal possible dimension, i.e. there exists no such function θ : A → R d {\displaystyle \theta :A\to \mathbb {R} ^{d}} with d < n {\displaystyle d<n} .
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