ChapterZero

There is one lesson I’ve been constantly relearning this semester: in analysis, once you get past measure theory, expect the concepts to be hopelessly abstract at first. The ones that aren’t are few and far between; most of the time, I’ve found myself asking after a particularly involved proof, why did we put ourselves through that? The only things you can do to pierce the veil of this abstractness are: flat out ask what the point of some new concept is, ask for meaningful applications or counterexamples for the theorems, find several good books that approach the same ideas from different angles. Pretty much, learning analysis has become like the process of learning topology: you see some pretty and elegant concepts, but for the most part, you’re cramming knowledge down your craw now that will do you good later. My latest struggle, initiated just today, is with topological vector spaces, or more precisely, locally convex topological vector spaces: why do we care that a space is a LCTVS?

Possibly relevant posts:

• Topological spaces (6/21/2003)
• Vector space projection methods (2/22/2006)
• Topology (4/28/2005)