10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
10. Commutative Banach algebras - Aarhus Universitet
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Quotients<br />
Let X be a <strong>Banach</strong> space and J ⊆ X a closed subspace. Then the<br />
quotient space X/J comes equipped with the norm<br />
x + J = inf {x − j : j ∈ J} ,<br />
called the quotient norm.<br />
The reader should check that this is indeed a norm on the quotient<br />
space. E.g the triangle inequality is established as follows:<br />
When x,y ∈ X and j,j ′ ∈ J,<br />
x + y + J ≤ x + y − j − j ′ ≤ x − j + y − j ′ .<br />
By taking the infimum over all j ′ ∈ J we conclude that<br />
x + y + J ≤ x − j + y + J;<br />
then we take the infimum over all j ∈ J to conclude that<br />
x + y + J ≤ x + J + y + J.<br />
In the following we will often write q(x) = x + J and consider<br />
q : X → X/J as a linear surjection - the quotient map. Note that<br />
q(x) ≤ x for all x ∈ X.<br />
Klaus Thomsen <strong>10.</strong> <strong>Commutative</strong> <strong>Banach</strong> <strong>algebras</strong>