To solve variational indefinite problems, a celebrated tool is the Banach-Nečas- Babuška theory, which relies on the inf-sup condition. Here, we choose an al- ternate theory, T-coercivity. This theory uses explicit inf-sup operators, both at the continuous and discrete levels. It is applied to solve Helmholtz-like problems in acoustics and electromagnetics. In the process, we provide new and simpler proofs to solve the exact and discrete problems, and to show convergence. We also establish sharp estimates on the convergence rates.