An infinite dimensional version of the intermediate value theorem

Let $\f = I-k$ be a compact vector field of class $C^1$ 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 $\R^2$) and Kronecker (in $\R^k$), 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: \textsl{If a half-line with extreme $q \notin \f(\S)$ intersects transversally the function $\f|_\S$ for only one point of\, $\S$, then any value of the connected component of $\H\setminus\f(\S)$ containing $q$ is assumed by $\f$ in $\B$. In particular, such a component is bounded.}