An infinite dimensional version of the intermediate value theorem

Abstract

Let $$\mathfrak {f}= I-k$$ f = I - k be a compact vector field of class $$C^1$$ C 1 on a real Hilbert space $$\mathbb {H}$$ H . In the spirit of Bolzano’s Theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in $$\mathbb {R}^2$$ R 2 ) and Kronecker (in $$\mathbb {R}^k$$ R k ), we prove an existence result for the zeros of $$\mathfrak {f}$$ f in the open unit ball $$\mathbb {B}$$ B of $$\mathbb {H}$$ H . Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction $$\mathfrak {f}|_\mathbb {S}$$ f | S of $$\mathfrak {f}$$ f to the boundary $$\mathbb {S}$$ S of $$\mathbb {B}$$ B . As an extension of this, but not as a consequence, we obtain as well an Intermediate Value Theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical Intermediate Value Theorem: If a half-line with extreme$$q \notin \mathfrak {f}(\mathbb {S})$$ q ∉ f ( S ) intersects transversally the function$$\mathfrak {f}|_\mathbb {S}$$ f | S for only one point of $$\mathbb {S}$$ S , then any value of the connected component of$$\mathbb {H}{\setminus }\mathfrak {f}(\mathbb {S})$$ H \ f ( S ) containingqis assumed by$$\mathfrak {f}$$ f in$$\mathbb {B}$$ B . In particular, such a component is bounded.