пример summary по математике - саммэри английский - математический факультет функциональный анализ

Chapter 2

In second chapter we can read about completeness.
The validity of many important theorems of analysis depends on the completeness of the systems with which they deal. This accounts for the inadequacy of the rational number system and of the Riemann integral (to mention just the two best-known examples) and for the success encountered by their replacements, the real numbers and the Lebesgue integral.
Also we can read about Baire's theorem which is about complete metric spaces (often called the category theorem) and Banach-Steinhaus theorem about topological vector spaces.
Open mappings theorem shows us some additional properties of relations between topological vector spaces and F-spaces.
And finally we can read about the closed graph theorem and understand conditions of existence of graph closure.
In order to emphasize the role played by the concept of category, some theorems of this chapter are stated in a little more generality than is usually needed.
When this is done, simpler versions (more easily remembered but sufficient for most applications) are also given.

Chapter 3
This chapter deals primarily (though not exclusively) with the most important class of topological vector spaces, namely, the locally convex ones. The highlights, from the theoretical as well as the applied standpoints, are
• the Hahn-Banach theorems (assuring a supply of continuous linear functionals that is adequate for a highly developed duality theory)
• the Banach-Alaoglu compactness theorem in dual spaces
• the Krein­Milman theorem about extreme points.

In this chapter we can also read about weak and strong properties and relations between them. For compact convex sets concept of polar is also given.

Chapter 4

In this chapter we can read about DUALITY IN BANACH SPACES. Firstly we can find the definition of duality, there are also several theorems about duality in Banach spaces. The definition of weak topology of dual spaces is also used.
There’s information about the spectrum of operator – and some examples.
The main objective of the rest of this chapter is to analyze the spectrum of a compact operator.
Adjoints will play an important role in this investigation.

In the middle of this chapter we can read about annihilators and duals of subspaces and of quotient spaces.
Finally adjoints and compact operators are presented.

Thus reading this chapter we can understand the most important theorems and definitions related to duality in Banach spaces.