Measure, Integration & Real Analysis, 2021a

Chapter 6
Banach Spaces
We begin this chapter with a quick review of the essentials of metric spaces. Then
we extend our results on measurable functions and integration to complex-valued
functions. After that, we rapidly review the framework of vector spaces, which
allows us to consider natural collections of measurable functions that are closed under
addition and scalar multiplication.
Normed vector spaces and Banach spaces, which are introduced in the third section
of this chapter, play a hugely important role in modern analysis. Most interest focuses
on linear maps on these vector spaces. Key results about linear maps that we develop
in this chapter include the Hahn–Banach Theorem, the Open Mapping Theorem, the
Closed Graph Theorem, and the Principle of Uniform Boundedness.
Market square in Lwów, a city that has been in several countries because of changing
international boundaries. Before World War I, Lwów was in Austria–Hungary.
During the period between World War I and World War II, Lwów was in Poland.
During this time, mathematicians in Lwów, particularly Stefan Banach (1892–1945)
and his colleagues, developed the basic results of modern functional analysis. After
World War II, Lwów was in the USSR. Now Lwów is in Ukraine and is called Lviv.
CC-BY-SA Petar Milošević
Measure, Integration & Real Analysis, by Sheldon Axler
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