The injective (or direct) limit of C*-algebras is one way to construct new C*-algebras from directed system of C*-algebras (defined below), and is an essential tool in operator theory, so one may as well get acquainted with it. The projective limit (or inverse limit) is not as common it seems, but I will add it here for completeness. In this post I will try to give a definition of the construct by universal properties of colimits in the category of C*-algebras, but reducing the prerequisites from category theory to a bare minimum. The point is to highlight that similarities between direct limits of groups, rings, algebras etc., stems from the fact that they all solve the same universal problem in their respective categories, and to justify why some of these limits/colimits are preserved under certain transformations. Though the similarities may be evidenced, this is understandably (but also unfortunately) often not addressed in the classical references of operator theory, as a formal definition of a limit/colimit would be a significant digression.
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A tour of functional analysis 1 – Locally convex vector spaces and the Hahn-Banach theorem(s)
One of the pillars of functional analysis is the Hahn-Banach theorem, so it makes sense to dedicate a post to this theorem. On normed spaces, the theorem has a plethora of interesting corollaries, some of which will be stated here. The locally convex spaces are of interest since they are the most rudimentary topological vector spaces on which the the Hahn-Banach theorem can be used to extend continuous linear functionals, and encompasses a sizable chunk of the topological vector spaces one might meet in the wild.