We study some interesting characterizations of real inner product spaces expressed in terms of angular distances. We first discuss the equivalence of characterizing an inner product space via the usual angular distance and the p-angular distance. Then, we establish a parametric family of upper bounds for the usual angular distance which also serves as a characterization of an inner product space. As an application, bounds for the usual angular distance are utilized in obtaining improvements of the real Cauchy–Schwarz inequality. Finally, we give several comparative relations for angular distances in inner product spaces.

中文翻译：

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通过角距离和柯西-施瓦兹不等式表征内积空间

我们研究了一些真实的内积空间的有趣特征，这些特征用角距离表示。我们首先讨论通过通常的角距离和p角距离表征内积空间的等价性。然后，我们为通常的角距离建立了一个上界的参数族，这也可以作为内积空间的表征。作为一种应用，通常的角距离范围可用于获得实际柯西-施瓦兹不等式的改进。最后，我们给出内积空间中角距离的几个比较关系。