Normed spaces and
The objects in functional analysis are function spaces endowed with topolo-
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
with the uniform convergence, defined by the condition
f − fn
|f(t) − fn(t)| → 0.
Examples of operators on this space are the integral operators
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
K(x, y)||f(y) − fn(y)| dy ≤ M f − fn L,
so that T satisfies the continuity condition Tf −Tfn
→ 0 if f −fn
Note that if L = [a, b], this space is infinite-dimensional, since it contains
the linearly independent functions 1, x,
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.
as well as the operators between that make the operations continuous