Basic Analysis – Gently Done Topological Vector Spaces

Next, we consider the triangle inequality;
�[x]+[y]� = �[x+y]�
= inf �x+y +m�
m∈M
= inf
m,m ′ ∈M �x+m+y +m′ �
≤ inf
m,m ′ ∈M (�x+m�+�y +m′ �)
= �x�+�y�.
8: Banach Spaces 89
Clearly, �[x]� ≥ 0 and, as noted already, �[0]� = 0, so � ·� is a seminorm on
the quotient space X/M. To see whether or not it is a norm, all that remains is
the investigation of the implication of the equality �[x]� = 0. Does this imply that
[x] = 0 in X/M? We will see that, in general, the answer is no, but if M is closed
the answer is yes, as the following argument shows.
Proposition 8.5 Suppose that M is a closed linear subspace of the normed space
X. Then �·� as defined above is a norm on the quotient space X/M—called the
quotient norm.
Proof According to the discussion above, all that we need to show is that if x ∈ X
satisfies �[x]� = 0, then [x] = 0 in X/M, that is, x ∈ M.
So suppose that x ∈ X and that �[x]� = 0. Then infm∈M �x + m� = 0, and
hence, for each n ∈ N, there is zn ∈ M such that �x+z n� < 1
n . This means that
−zn → x in X as n → ∞. Since M is a closed subspace, it follows that x ∈ M
and hence [x] = 0 in X/M, as required.
Proposition 8.6 Let M be a closed linear subspace of a normed space X and let
π : X → X/M be the canonical map π(x) = [x], x ∈ X. Then π is continuous.
Proof Suppose that x n → x in X. Then
and the result follows.
�π(x n )−π(x)� = �[x n ]−[x]�
= �[x n −x]�
= inf
m∈M �x n −x+m�
≤ �x n −x�, since 0 ∈ M,
→ 0 as n → ∞