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 extreme \(q \notin \mathfrak {f}(\mathbb {S})\) intersects transversally the function \(\mathfrak {f}|_\mathbb {S}\) for only one point of \(\mathbb {S}\), then any value of the connected component of \(\mathbb {H}{\setminus }\mathfrak {f}(\mathbb {S})\) containing q is assumed by \(\mathfrak {f}\) in \(\mathbb {B}\). In particular, such a component is bounded.

令\(\mathfrak {f}= Ik\)为实希尔伯特空间\(\mathbb {H}\)上的类\(C^1\ )的紧致向量场。本着关于有界实数区间中存在零点的博尔扎诺定理的精神，以及柯西（在\(\mathbb {R}^2\)中）和克罗内克（在\(\mathbb {R}中）的扩展^k\) ），我们证明\(\mathbb {H}\) 的开单位球\(\mathbb {B}\)中\(\mathfrak {f}\)的零点存在。与经典的有限维结果类似，零点的存在完全是从\(\mathfrak {f}\)的限制 \(\mathfrak {f}|_\mathbb {S}\) 推导出来的到\(\mathbb {B}\)的边界\(\mathbb {S}\)。作为其扩展，但不是结果，我们还获得了一个中值定理，其陈述需要拓扑度。这样的结果意味着以下易于理解的、非平凡的经典中值定理的推广：如果具有极值\(q \notin \mathfrak {f}(\mathbb {S})\) 的半线 横向相交函数\ (\mathfrak {f}|_\mathbb {S}\)仅表示\(\mathbb {S}\)的一个点，则\(\mathbb {H}{\setminus }\)的连通分量的任意值mathfrak {f}(\mathbb {S})\)包含q假设为 \(\mathfrak {f}\) 中的 \(\mathbb {B}\)。特别地，这样的组件是有界的。