  The $L^2$-orthogonal projection onto a subspace is an important mathematical
tool, which has been widely applied in many fields such as linear least squares
problems, eigenvalue problems, ill-posed problems, and randomized algorithms.
In some numerical applications, the entries of a matrix will seldom be known
exactly, so it is necessary to develop some bounds to characterize the effects
of the uncertainties caused by matrix perturbation. In this paper, we establish
new perturbation bounds for the $L^2$-orthogonal projection onto the column
space of a matrix, which involve upper (lower) bounds and combined upper
(lower) bounds. The new bounds contain some sharper counterparts of the
existing ones. Numerical examples are also given to illustrate our theoretical
results.
