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