MATH 675 HOMEWORK 5
SOLUTIONS
Exercise 1. Show that the operator norm satisfies all of the properties of a norm.
Let T L(X, Y ). Then kT k = sup
x6=0
kT xkY
. We have to prove three things:
kxkX
(a) kT k 0 and kT k = 0 if and only if T = 0. Since for all
limit superior
and
limit inferior
Ideas/Defs
Let cfw_sn
n=1 be a sequence in R. Define two corresponding sequences cfw_ak k=1 and cfw_bk k=1 , which examine
the behavior of the tails of cfw_sn
n=1 , by
ak = inf sn
= glbcfw_sn : n k
R cfw_
an increasing
2
The Classical Sequence Spaces
We now turn to the classical sequence spaces p for 1 p < and c0 .
The techniques developed in the previous chapter will prove very useful in
this context. These Banach spaces are, in a sense, the simplest of all Banach
spac
THE CONTRACTION MAPPING THEOREM
KEITH CONRAD
1. Introduction
Let f : X X be a mapping from a set X to itself. We call a point x X a fixed point
of f if f (x) = x. For example, if [a, b] is a closed interval then any continuous function
f : [a, b] [a, b] h
Problem 1 (15 points) Let c0 be the space consisting of all sequences that converge to 0
equipped with the supremum norm. Let l be the space consisting of all bounded sequences
also equipped with the supremum norm.
(a) Show that c0 is a closed subspace of
2
Linear Operators on Normed Spaces
Many of the basic problems of applied mathematics share the property of
linearity, and linear spaces and linear operators provide a general and useful
framework for the analysis of such problems. More complicated applic