Basic Analysis – Gently Done Topological Vector Spaces

8: Banach Spaces 91
for any t,s ∈ C, and f,g ∈ X. Hence φ : X/M → C is linear. Furthermore,
φ([f]) = φ([g]) ⇐⇒ f(0) = g(0)
⇐⇒ f ∼ g
⇐⇒ [f] = [g]
and so we see that φ is one-one.
Given any s ∈ C, there is f ∈ X with f(0) = s and so φ([f]) = s. Thus φ is
onto. Hence φ is a vector space isomorphism between X/M and C, i.e., X/M ∼ = C
as vector spaces.
Now, it is easily seen that M is closed in X with respect to the � · � ∞ -norm
and so X/M is a Banach space when given the quotient norm. We have
�[f]� = inf{�g� ∞ : g ∈ [f]}
= inf{�g� ∞ : g(0) = f(0)}
= |f(0)| (take g(s) = f(0) for all s ∈ [0,1]).
That is, �[f]� = |φ([f])|, for [f] ∈ X/M, and so φ preserves the norm. Hence
X/M ∼ = C as Banach spaces.
Now consider X equipped with the norm �·� 1 . Then M is no longer closed in
X. We can see this by considering, for example, the sequence (gn ) given by
�
ns, 0 ≤ s ≤ 1/n
gn (s) =
1, 1/n < s ≤ 1.
Then g n ∈ M, for each n ∈ N, and g n → 1 with respect to �·� 1 , but 1 /∈ M. The
‘quotient norm’ is not a norm in this case. Indeed, �[f]� = 0 for all [f] ∈ X/M.
To see this, let f ∈ X, and, for n ∈ N, set h n (s) = f(0)(1−g n (s)), with g n defined
as above. Then h n (0) = f(0) and �h n � 1 = |f(0)|/2n. Hence
which implies that
inf{�g� 1 : g(0) = f(0)} ≤ �h n � 1 ≤ |f(0)|/2n
�[f]� = inf{�g� 1 : g ∈ [f]} = 0.
The ‘quotient norm’ on X/M assigns ‘norm’ zero to all vectors.
Next we discuss linear mappings between normed spaces. These are also called
linear operators.
Definition 8.9 The linear operator T : X → Y from a normed space X into a
normed space Y is said to be bounded if there is some k > 0 such that
�Tx� ≤ k�x�
for all x ∈ X. If T is bounded, we define �T� to be
�T� = inf{k : �Tx� ≤ k�x�, x ∈ X}.
We will see shortly that � · � really is a norm on the set of bounded linear
operatorsfrom X into Y. The following result follows directlyfrom thedefinitions.