Banach Spaces and Fixed-Point Theorems

Abstract

In a Banach space, the so-called norm

$$ \parallel u\parallel = nonnegativenumber \hfill \\ $$

is assigned to each element u. This generalizes the absolute value |u of a real number u. The norm can be used in order to define the convergence

$$ \mathop {\lim }\limits_{n \to \infty } {u_n} = u \hfill \\ $$

by means of

$$ \mathop {\lim }\limits_{n \to \infty } \parallel {u_n} - u\parallel = 0. \hfill \\ \parallel u\parallel = nonnegativenumber \hfill \\ $$

The role of functional analysis has been decisive exactly in connection with classical problems. Almost all problems are on the applications, where functional analysis enables one to focus on a specific set of concrete analytical tasks and organize material in a clear and transparent form so that you know what the difficulties are.

Concrete and functional analysis exist today in an inextricable symbiosis. When someone writes down a system of axioms, no one is going to take them seriously, unless they arise from some intuitive body of concrete subject matter that you would really want to study, and about which you really want to find out something. Felix E. Browder, 1975