Single-walled carbon nanotubes provide an ideal system for studying the properties of one-dimensional (1D) materials, where strong electron-electron interactions are expected. Optical measurements have recently reported the existence of excitons in semiconducting nanotubes, revealing the importance of many-body effects. Surprisingly, pioneering electronic structure calculations and scanning tunnelling spectroscopy (STS) experiments report the same gap values as optical experiments. Here, an experimental STS study of the bandgap of single-walled semiconducting nanotubes, demonstrates a continuous transition from the gap reduced by the screening resulting from the metal substrate to the intrinsic gap dominated by many-body interactions. These results provide a deeper knowledge of many-body interactions in these 1D systems and a better understanding of their electronic properties, which is a prerequisite for any application of nanotubes in the ultimate device miniaturization for molecular electronics, or spintronics.