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Let $X$ be a compact Hausdorff space and let $C(X)$ denote the set of continuous complex valued functions on $X$. Define $$ \|f\|:=\sup\{|f(x)|:x\in X\},$$

then prove that $\|fg\|\leq \|f\|\|g\|$.

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What are you struggling with? Do you understand the definition of the norm? –  Michael Albanese Jul 7 at 8:04
    
yes i know the definition but i do not know why it is inequality,it should be equality.But now i understand that how this is inequality.Thanks to everyone. –  Nannes Jul 7 at 9:06
    
In future, you should include such concerns in the post so people can address them. –  Michael Albanese Jul 7 at 9:10

3 Answers 3

up vote 4 down vote accepted

For every $x \in X $ we have $|g(x)| \leq \sup_{z \in X} |g(z)|$ by definition of the supremum, so for every $x \in X$ we may observe that $$|f(x)g(x)| = |f(x)||g(x)|\leq |f(x)|\left(\sup_{z \in X}|g(z)|\right) =|f(x)|\|g\|,$$ Since this is true for every $x\in X$ we may take the supremum on both sides of the equation to get $$\|fg\| = \sup_{x \in X}|f(x)g(x)| \leq \sup_{x \in X} |f(x)|\|g\| = \|f\|\|g\|$$

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yes perfect.Thanx –  Nannes Jul 7 at 9:08

Alternatively, as $\{(x,x)| x\in X\} \subset \{(x,y)| x,y\in X\}$: $$ \sup_{x\in X} |f(x)g(x)| \le \sup_{x,y\in X} |f(x)g(y)| = \sup_{x\in X}|f(x)| \sup_{y\in X}|g(y)| $$

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How do you justify $\sup_{x \in X} |f(x)g(x)| \leq \sup_{x,y \in X} |f(x)g(y)|$ ? –  Surb Jul 7 at 8:27
    
this is because $\{(x,x) | x\in X\}\subset X\times X$. –  mookid Jul 7 at 8:49
    
ok, it is really alternative then :). –  Surb Jul 7 at 8:52

For all $x \in X$ we have $$|f(x)| \leq \|f\| ,$$ as well as $$|g(x)| \leq \|g\|. $$ Multiplying the two inequalities gives

$$|f(x)g(x)| \leq \|f\|\|g\| \; \forall x \in X .$$

Thus $\|f\| \|g\|$ is an upper bound of the set $\{|f(x)g(x)|:x \in X\}$, hence the least upper bound can't be larger.

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