Linear Analysis Part II

The objects in this course are infinite dimensional vector spaces (hence the term “linear”) over R or C, together with additional structure (a “norm” or “inner product”) which “respects” in some way the linear structure. This additional structure will allow us to do “analysis”. The most pedestrian way to understand the last sentence is that it will allow us to “take limits”. In fact, the extra structure allows much more than just “taking limits”. Hidden in the notion of norm and inner product are notions of convexity, duality, and orthogonality. In the most complicated structure we will discuss, that of a Hilbert space, all these notions will be present simultaneously together with completeness. The point of view of this course, like most undergraduate mathematics courses, will be axiomatic/revisionist. The objects will be defined, and the basic theorems discussed without any “motivation”. The objects in this course have been come to be viewed as so central for mathematics that it is presently inaccurate to consider them as having any single “motivation”. This being said, the subject arose and developed under specific mathematical circumstances, and this may be useful for deeper understanding of the theory. Nice notes can be found in [3, 2]. At the very least, however, everyone starting out in this subject should know that the vector spaces that originally gave rise to this theory were spaces of functions, and the linear operators were linear partial differential operators, their inverses, and more generally so-called linear integral operators. So the linear algebraic question “Under what conditions can you invert an operator?”–and this represents the kind of question we will be answering in this course–in practice meant “Under what conditions can you solve a linear pde?”. Because of this connection to spaces of functions, the subject is almost always known by the name Functional Analysis, and it is under this name that you will probably search the literature for more material, if you find these notes to be confusing.