Auerbach's Theorem

Auerbach's Theorem states that in any infinite-dimensional Banach space, there exists a special set called a biorthogonal system, which consists of pairs of vectors and corresponding linear functionals. These sets help us understand the structure of the space by providing a way to "coordinate" elements within it. Essentially, the theorem guarantees that even in complex, infinite-dimensional spaces, we can find collections that behave like coordinate axes, aiding in analysis and approximation. This result is fundamental in functional analysis, bridging the gap between abstract spaces and more manageable, structured systems.