A non compactness measure with values in the lattice of extended non negative real numbers $$[0, +\infty ]$$ is introduced in the general setting of a Hausdorff topological vector space E. This generalizes the classical Kuratowski and Hausdorff non compactness measures. In order to achieve this, we introduce the notions of basic and sufficient collections of zero neighborhoods. We then show that our measure satisfies most of the properties of the classical non compactness measures. We particularly show that if E is locally p-convex for some $$0 < p \le 1$$, our measure is stable by the transition to the closed p-convex hull. This allows us to obtain, as applications, generalizations of the well-known three fixed point theorems, namely Schauder, Darbo, and Sadovskii’s ones in the setting of locally p-convex spaces. As another application, we establish a quantification of Ascoli theorem in the space C(X, E) of vector-valued continuous functions on a Hausdorff completely regular space with values in a topological vector space E, giving an alternative of Ambrosetti theorem initially stated in the metric spaces setting.

中文翻译：

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拓扑向量空间和应用中的实值非紧性度量

在 Hausdorff 拓扑向量空间 E 的一般设置中引入了在扩展非负实数 $$[0, +\infty ]$$ 的格中具有值的非紧性度量。 这概括了经典的 Kuratowski 和 Hausdorff 非紧性度量. 为了实现这一点，我们引入了零邻域的基本和充分集合的概念。然后我们证明我们的度量满足经典非紧凑性度量的大多数属性。我们特别表明，如果 E 在 $$0 < p \le 1$$ 的情况下是局部 p 凸的，则我们的度量通过向封闭 p 凸包的过渡而稳定。这使我们能够将著名的三个不动点定理（即 Schauder、Darbo 和 Sadovskii 的定理）在局部 p 凸空间的设置中作为应用得到推广。作为另一个应用程序，