O'Neil's Theorem is a result in the field of functional analysis, specifically concerning the behavior of certain types of linear operators. It provides conditions under which a bounded linear operator can be approximated by finite-rank operators, which are simpler to analyze and compute. This theorem is particularly useful in the study of Hilbert spaces and Banach spaces. The theorem is named after William O'Neil, who contributed to the understanding of operator theory. O'Neil's Theorem helps mathematicians and scientists understand how complex systems can be simplified, making it easier to solve problems in various applications, including quantum mechanics and signal processing.