Hahn-Banach Theorem

The Hahn-Banach Theorem is a fundamental principle in mathematics that allows us to extend a linear function defined on a smaller part of a space to the entire space without losing certain properties, especially its size or "boundedness." In simple terms, if you have a way to measure or evaluate objects within a limited context, this theorem guarantees you can expand that measurement consistently across the whole space, preserving its essential features. It’s a powerful tool used in functional analysis and optimization, ensuring flexibility in analyzing and working with functions in infinite-dimensional spaces.