The spaces of continuous functions on a compact Hausdorff space $K$ are some of the most familiar Banach spaces yet much about their structure remains unknown. In this talk we will review some of what is known about the real-valued case and a basic open question on projections. We will also mention some recent results about the Banach space-valued case.

The talk will not assume knowledge of Banach space theory but will require some basic functional analysis like that found in a course in real analysis. Some familiarity with ordinals and their arithmetic will also be helpful.