In this thesis, we present an expanded account of inequalities for compact operators on Hilbert spaces based on an article by Erlijman, Farenick and Zeng (2001). We study some formulation of the fundamental real-number inequalities for compact operators. Although there is an extensive body of works concerning operators inequalities, especially those that arise from operator-monoton or operator-convex functions, there are far fewer result concerning operator inequalities that arise through spectral or singular values inequalities. Thompson (1978) proved the first fundamental inequality namely the triangle inequality for the $nimes n$ complex matrices. Thompson's work was later generalized to the case of arbitrary von-Neumman algebras by Akemman, Anderson and Pederson (1982).