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A homework problem:

Let $\phi:X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces. Let $\lambda$ be a continuous linear functional on $C(Y)$.
Show that there is a continuous linear functional $\theta$ on $C(X)$ such that:

  1. $\|\theta\|=\|\lambda\|$
  2. $\theta(f \circ \phi)=\lambda(f)$ and any $f \in C(Y)$

My attempt: I am stuck about in the beginning. I read about the Hahn-Banach theorems in Rudin's functional analysis. Those theorems seem relevant because they talk about existence of functionals, but there are several versions and I'm confused about which of them to use, if any (I am allowed to quote this book). I think maybe the second condition can tell me how to define a functional on some subspae of $C(X)$ and then use an "extension theorem" (that's how some of those theorems are refered to in the book), but that's just general intuition.

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up vote 1 down vote accepted

Hints:

1) Show that $E=\{f\circ \varphi:f\in C(Y)\}$ is linear subsapce of $C(Y)$.

2) Show that $f_1\circ\varphi=f_2\circ\varphi\implies f_1=f_2$

3) Using 2) show that you have well defined functional $\theta_0:E\to\mathbb{C}:g\mapsto \lambda(f)$ where $g=f\circ\varphi$.

4) Show that $\Vert f\circ\varphi\Vert=\Vert f\Vert$

5) Using 4) prove that $\Vert\theta_0\Vert=\Vert\lambda\Vert$

6) Using Hahn-Banach get norm preserving extenesion $\theta$ of $\theta_0$. This will be the desired functional

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