An infinite dimensional version of the intermediate value theorem

Abstract

Let mathfrak {f}= I-k be a compact vector field of class C^1 on a real Hilbert space mathbb {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) and Kronecker (in mathbb {R}^k), we prove an existence result for the zeros of mathfrak {f} in the open unit ball mathbb {B} of mathbb {H}. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction mathfrak {f}|_mathbb {S} of mathfrak {f} to the boundary mathbb {S} of mathbb {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 extremeq notin mathfrak {f}(mathbb {S})intersects transversally the functionmathfrak {f}|_mathbb {S}for only one point of mathbb {S}, then any value of the connected component ofmathbb {H}{setminus }mathfrak {f}(mathbb {S})containingqis assumed bymathfrak {f}inmathbb {B}. In particular, such a component is bounded.