Let $X$ be a compact metric space and let $F$ be a closed subset of $X$. If $W = \{f \in C(X) : f(x) = 0 \text{ for all } x \in F\}$, show that $C(X)/W$ is isometrically isomorphic to $C(F)$. ($C(X)$ is the space of continuous real-valued functions on $X$.)
Quotient space is isometrically isomorphic to $C(F)$
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functional-analysis
banach-spaces
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0The comment here http://math.stackexchange.com/questions/123133/hahn-banach-theorem#comment284673_123133 seems relevant – 2012-03-25
1 Answers
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Hint: Consider the obvious map $C(X) \to C(F)$ (restriction). Prove that it is a quotient map (i. e. maps the open unit ball onto the open unit ball) and determine its kernel ...