Teorema de Hahn-Banach

The Hahn-Banach Theorem is a fundamental result in functional analysis stating that continuous linear functions (called linear functionals) defined on a smaller part of a space can be extended to the entire space without losing their properties or increasing their size. In simpler terms, if you have a way to measure or evaluate a small subset of a space, you can always extend that measurement to the whole space in a consistent way, ensuring the extension remains controlled and doesn't "blow up" in size. This theorem underpins many results in mathematics, especially in understanding the structure of function spaces.