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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, bounded functions are functions for which there exists a lower bound and an upper bound, in other words, a constant which is larger than the absolute value of any value of this function. If we consider a family of bounded functions, this constant can vary between functions. If it is possible to find one constant which bounds all functions, this family of functions is uniformly bounded.The uniform boundedness principle in functional analysis provides sufficient conditions for uniform boundedness of a family of operators.