Basic Analysis – Gently Done Topological Vector Spaces

90 Basic Analysis
Proposition 8.7 For any closed linear subspace M of a Banach space X, the
quotient space X/M is a Banach space under the quotient norm.
Proof We know that X/M is a normed space, so all that remains is to show
that it is complete. We use the criterion established above. Suppose, then, that
([xn ]) is any sequence in X/M such that �
n�[xn ]� < ∞. We show that there is
[y] ∈ X/M such that �k n=1 [xn ] → [y] as k → ∞.
For each n, �[xn ]� = infm∈M �xn + m�, and therefore there is mn ∈ M such
that
�xn +mn� ≤ �[xn ]�+ 1
,
2n by definition of the infimum. Hence
�
�xn +mn� ≤ ��
�[xn ]�+ 1
2n �
n
n
< ∞.
But(x n +m n )isasequenceintheBanachspaceX, andsolim k→∞
exists in X. Denote this limit by y. Then we have
�
�
k�
[xn ]−[y] � �
k�
� = � [xn −y] � �
n=1
n=1
�
k�
= inf � xn −y +m � �
≤ � �
= � �
m∈M
n=1
k�
xn −y +
n=1
k�
n=1
�
m �
n
k�
(xn +mn )−y � �
n=1
→ 0, as k → ∞.
� k
n=1 (x n +m n )
Hence �k n=1 [xn ] → [y] as k → ∞ and we conclude that X/M is complete.
Example 8.8 Let X be the linear space C([0,1]) and let M be the subset of X
consisting of those functions which vanish at the point 0 in [0,1]. Then M is a
linear subspace of X and so X/M is a vector space.
Define themapφ : X/M → Cbysettingφ([f]) = f(0), for[f] ∈ X/M. Clearly,
φ is well-defined (if f ∼ g then f(0) = g(0)) and we have
φ(t[f]+s[g]) = φ([tf +sg])
= tf(0)+sg(0)
= tφ([f])+sφ([g])
Department of Mathematics King’s College, London