Bounded operators are mathematical functions that map elements from one vector space to another while maintaining a limit on how much they can stretch or compress those elements. In simpler terms, if you apply a bounded operator to a vector, the result will not be excessively larger than the original vector, ensuring that the operator behaves predictably. These operators are crucial in functional analysis, particularly in the study of Hilbert spaces and Banach spaces. They allow for the extension of concepts from finite-dimensional spaces to infinite-dimensional ones, making them essential for understanding various applications in quantum mechanics and signal processing.