We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.

翻译：我们从由Sz\'{e}kely、Rizzo和Bakirov引入的Euclidean空间的距离(Brownian)共变理论扩大到一般的公制空间。我们证明,为了测试独立性,衡量空间必须而且足够为强烈的负型。特别是,我们表明,这可以支持分离的Hilbert空间,该空间回答了Kosorok的问题。我们使用的是Fourier变换的操纵,而不是在Hilbert空间使用基本不平等的衡量空间和嵌入。