An infinite dimensional version of the intermediate value theorem

Let f = I - k be a compact vector field of class C1 on a real Hilbert space 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 R2) and Kronecker (in Rk), we prove an existence result for the zeros of f in the open unit ball B of H. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction f|S of f to the boundary S of 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 ?/ f(S) intersects transversally the function f|S for only one point of S, then any value of the connected component of H\f(S) containing q is assumed by f in B. In particular, such a component is bounded.