The electronic transport properties of conventional three-dimensional metals are successfully described by Fermi-liquid theory. But when the dimensionality of such a system is reduced to one, the Fermi-liquid state becomes unstable to Coulomb interactions, and the conduction electrons should instead behave according to Tomonaga-Luttinger-liquid (TLL) theory. Such a state reveals itself through interaction-dependent anomalous exponents in the correlation functions, density of states and momentum distribution of the electrons. Metallic single-walled carbon nanotubes (SWNTs) are considered to be ideal one-dimensional systems for realizing TLL states. Indeed, the results of transport measurements on metal-SWNT and SWNT-SWNT junctions have been attributed to the effects of tunnelling into or between TLLs, although there remains some ambiguity in these interpretations. Direct observations of the electronic states in SWNTs are therefore needed to resolve these uncertainties. Here we report angle-integrated photoemission measurements of SWNTs. Our results reveal an oscillation in the pi-electron density of states owing to one-dimensional van Hove singularities, confirming the one-dimensional nature of the valence band. The spectral function and intensities at the Fermi level both exhibit power-law behaviour (with almost identical exponents) in good agreement with theoretical predictions for the TLL state in SWNTs.