The Uniform Boundedness Principle

The Uniform Boundedness Principle states that if a collection of mathematical functions (operators) each acts on elements in a way that doesn’t become arbitrarily large for every input, then there must be a common limit to how large they can get across all inputs. In other words, if none of the functions blow up individually for all inputs, then there’s a universal bound that applies to all functions in the collection, ensuring their sizes are controlled uniformly. This principle is fundamental in analysis, particularly in studying infinite-dimensional spaces, as it links individual boundedness to a collective bound.