In various fields, observations are curves over some natural time interval. These curves often arise from finely spaced measurements (e.g., in physical sciences and finance) or from smoothing unequally spaced sparse measurements. Recent years have seen the development of tools for analyzing such data in the growing field of functional data analysis. To validate the assumptions underlying these tools, it is important to verify that the functional observations form a simple random sample. If the curves form a (functional) time series, then model validation typically relies on checking whether model residuals are independent and identically distributed. We propose a test for independence and identical distribution of functional observations. To reduce dimension, curves are projected on the most important functional principal components, then a test statistic based on lagged cross-covariances of the resulting vectors is constructed. We show that this dimension-reduction step introduces asymptotically negligible terms; that is, the projections behave asymptotically as iid vector–valued observations. A complete asymptotic theory based on correlations of random matrices, functional principal component expansions, and Hilbert space techniques is developed. The test statistic has a chi-squared asymptotic null distribution and can be readily computed using the R package fda. The test has good empirical size and power, which in our simulations and examples is not affected by the choice of the functional basis. Its application is illustrated on two data sets: credit card sales activity and geomagnetic records.