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.

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as well as the operators between that make the operations continuous

http://dx.doi.org/10.1090/gsm/116/02