Chapter 2
Normed spaces and
operators
The objects in functional analysis are function spaces endowed with topolo-
gies
them. This chapter is devoted to the most basic facts concerning Banach
spaces and bounded linear operators.
It can be useful for the reader to retain as a first model of function spaces
the linear space C(L) of all real continuous functions on a compact set L in
Rn
with the uniform convergence, defined by the condition
f fn
L
:= max
t∈L
|f(t) fn(t)| 0.
Examples of operators on this space are the integral operators
Tf(x) =
L
K(x, y)f(y) dy
where K(x, y) is continuous on L × L. Then T : C(L) C(L) is linear and
Tf Tfn
L
max
x∈L
L
K(x, y)||f(y) fn(y)| dy M f fn L,
so that T satisfies the continuity condition Tf −Tfn
L
0 if f −fn
L

0.
Note that if L = [a, b], this space is infinite-dimensional, since it contains
the linearly independent functions 1, x,
x2,
etc. Two major differences with
respect to the usual finite-dimensional Euclidean spaces are that a linear
map between general Banach spaces is not necessarily continuous and that
the closed balls are not compact.
25
as well as the operators between that make the operations continuous
http://dx.doi.org/10.1090/gsm/116/02
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