We have investigated the electron spin resonance at nonzero wave vector in GaAs single quantum wells by combining the virtues of high frequency surface acoustic wave generation to produce excitations with large wave numbers with a sensitive optical scheme to detect resonant absorption. The observed large deviations from the single particle Zeeman energy are attributed to the exchange interaction. The enhancement of the electronic g(*) factor is, however, substantially smaller compared with theoretical predictions for spin waves when adopting a bare Coulomb interaction potential.