The paper establishes several robustness, optimality, and convergence properties of the widely used class of instantaneous-gradient adaptive algorithms. The analysis is carried out in a purely deterministic framework and assumes no apriori statistical information. It starts with a simple Cauchy-Schwarz inequality for vectors in an Euclidean space and proceeds to derive local and global energy bounds that are shown here to highlight, as well as explain, several relevant aspects of this important class of algorithms.