Krasnoselskii genus

In nonlinear functional analysis, the Krasnoselskii genus generalizes the notion of dimension for vector spaces. The Krasnoselskii genus of a linear space is the smallest natural number for which there exists a continuous odd function of the form. 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

be a Banach space,
be the collection of symmetric closed subsets of,
the space of continuous functions.

For define the set
Then the Krasnoselskii genus of is defined as
In other words, if then there exists a continuous odd function such that. Moreover is the minimal possible dimension, i.e. there exists no such function with.

Properties

Let be a bounded symmetric neighborhood of in. Then the genus of its boundary is.
For, the following holds:

If there exists an odd function, then.
If, then.
If there exists an odd homeomorphism between and, then.

Combining these statements, it follows immediately that if there exists an odd homeomorphism between and then.