The Hahn–Banach theorem is one of the major theoremsproved in any first course on functional analysis. It has plentyof applications, not only within the subject itself, but alsoin other areas of mathematics like optimization, partial differentialequations, and so on. This article will give a briefoverview of the Hahn–Banach theorem, its ramifications andindicate some applications.
Two important results in linear algebra are the ‘rank-nullity theorem’ and the equality of the row and column ranks of a matrix. In this note, we will give a simple proof of the latter, using the former. As a by-product, we also prove the Fredholm alternative, which characterizes the range of the linear operator associated with a matrix.
