Ag/Pt(111) has been studied by spin-, angle-, and energy-resolved photoemission with circularly polarized synchrotron radiation of BESSY for different Ag coverages. The prepared layers were characterized by AES and LEED and turned out to grow epitaxially. An independent layer thickness determination and some additional structure information were obtained from Rutherford backscattering studies in combination with channeling. The spin-resolved photoemission experiments were performed for normal incidence of circularly polarized light and normal emission of the photoelectrons. The spin information allowed a separation of spin-orbit split peaks of the Ag adsorbate layer and partly also a separation of Ag adsorbate peaks from the Pt substrate background intensity. This method-induced improved resolution was used to deduce information about the development of the Ag band structure in the [Lambda]-direction from the coverage dependence of spin-resolved photoemission spectra. Dispersion of occupied bands was found to be already almost completely developed for a three-layer system. Information about the development of the band structure in its unoccupied part was obtained from the resonant behavior of peaks at about 6 eV below EF. The resonant behavior is already observed for three Ag layers and increases with layer thickness. We observe a peak in the photoemission spectra contributing to the resonances which is not predicted by existing theoretical models.