<p>As data becomes increasingly complex, measuring dependence among variables is of great interest. However, most existing measures of dependence are limited to the Euclidean setting and cannot effectively characterize the complex relationships. In this paper, we propose a novel method for constructing independence tests for random elements in Hilbert spaces, which includes functional data as a special case. Our approach is using distance covariance of random projections to build a test statistic that is computationally efficient and exhibits strong power performance. We prove the equivalence between testing for independence expressed on the original and the projected covariates, bridging the gap between measures of testing independence in Euclidean spaces and Hilbert spaces. Implementation of the test involves calibration by permutation and combining several <i>p</i>-values from different projections using the false discovery rate method. Simulation studies and real data examples illustrate the finite sample properties of the proposed method under a variety of scenarios.</p>