  We study the relativistic version of the $d$-dimensional isotropic quantum
harmonic oscillator based on the spinless Salpeter equation. This has no exact
analytical solutions. We use perturbation theory to obtain compact formulas for
the first and second-order relativistic corrections; they are expressed in
terms of two quantum numbers and the spatial dimension $d$. The formula for the
first-order correction is obtained using two different methods and we
illustrate how this correction splits the original energy into a number of
distinct levels each with their own degeneracy. Previous authors obtained
results in one and three dimensions and our general formulas reduce to them
when $d=1$ and $d=3$ respectively. Our two-dimensional results are novel and we
provide an example that illustrates why two dimensions is of physical interest.
We also obtain results for the two-dimensional case using a completely
independent method that employs ladder operators in polar coordinates. In
total, three methods are used in this work and the results all agree.
