Basic Analysis – Gently Done Topological Vector Spaces

8. Banach Spaces
In this chapter, we introduce Banach spaces and spaces of linear operators. Recall
that if X is a normed space with norm �·�, then the formula d(x,y) = �x−y�,
for x,y ∈ X, defines a metric d on X. Thus a normed space is naturally a metric
space and all metric space concepts are meaningful. For example, convergence of
sequences in X means convergence with respect to the above metric.
Definition 8.1 A complete normed space is called a Banach space.
Thus, a normed space X is a Banach space if every Cauchy sequence in X
converges—where X is given the metric space structure as outlined above. One
may consider real or complex Banach spaces depending, of course, on whether X
is a real or complex linear space.
Examples 8.2
1. If R is equipped with the norm �λ� = |λ|, λ ∈ R, then it becomes a real Banach
space. More generally, for x = (x 1 ,x 2 ,...,x n ) ∈ R n , define
�
�n
�x� =
i=1
|xi | 2
�1/2 Then, with this norm, R n becomes a real Banach space (when equipped with the
obvious component-wise linear structure).
In a similar way, one sees that C n , equipped with the similar norm, is a (complex)
Banach space. These norms are the Euclidean norms on R n and C n , respectively.
2. Equip C([0,1]), the linear space of continuous complex-valued functions on the
interval [0,1], with the norm
�f� = sup{|f(x)| : x ∈ [0,1]}.
Then C([0,1]) becomes a Banach space. This norm is called the supremum (or
uniform) norm and is often denoted �·� ∞ . Notice that convergence with respect
to this norm is precisely that of uniform convergence of the functions on [0,1].
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