Evidence for modified electronic structure in double-walled carbon nanotubes with respect to their individual inner and outer constituent single-walled nanotubes is provided by scanning tunneling microscopy and spectroscopy experiments. The contribution originating from the inner tube to the local density of states of the double-walled system was identified in agreement with previous theoretical calculations. Consequently, the chiral index for the inner tube was extracted based on the additional van Hove singularities present in the experimental tunneling spectra.