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For a compact metric space $X$, $C(X)$ denotes the space of continuous real-valued functions on $X$ equipped with the supremum norm. Let $X$ and $Y$ be compact metric space and let $g:X \to Y$ be a continuous map. Define $T: C(Y) \to C(X)$ by $T(f) = f\circ g$ . Clearly, $T$ is a linear transformation.

  1. Prove that $T$ is bounded. What is the value of $\|T\|$?
  2. Give a necessary and sufficient condition on $g$ for $T$ to be onto.
  3. Give a necessary and sufficient condition on $g$ for $T$ to be one-to-one.
  4. Give a necessary and sufficient condition on $g$ for $T$ to be an isometry.
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    What is your question? What did you try and where did you get stuck?2012-03-21
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    Hey @Chung, why don't you tell us where you're stuck and let us help you do this homework? I'm sorry you got 4 downvotes, I don't think that's deserved. The downvoters might remove the downvotes if you post your attempts and show us where you got stuck.2012-03-24
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    Why don`t you give a hints instead of him giving what he tried? Maybe some hints will be useful here.2012-03-25
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    It would be nice if people asked for help rather than just demanding answers...2012-03-25
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    I feel @t.b. 's comment here http://math.stackexchange.com/questions/123133/hahn-banach-theorem#comment284673_123133 still applies2012-03-25
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    More to the point: as a teacher, being told "Sir I am stuck" is next to useless. Being told "I don't understand what I have to prove" or "I can see how to do this in a special case, like this, but I don't know how to generalize" allows some actual *education* to be attempted2012-03-25
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    Are you @Danny?2012-03-27
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    @MattN. I'm afraid I disagree, and writing out a detailed answer as you have done below seems (to me) wasted effort until the OP takes note of http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question2012-03-27

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