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One theorem in Rudin's Real and complex analysis says the following:

If $X$ is a normed linear space and if $x_0 \in X$, $x_0\neq 0$, there is a bounded linear funcitonal $f$ on $X$, of norm 1, so that $f(x_0)=||x_0||$.

He uses the Hahn-Banach theorem to prove it, but why we cannot just say that $f(x)=||x||$ works?

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$f(x)=\|x\|$ is not linear!

There is no guarantee that $f(x+y) = \|x+y\|$ is equal to $f(x)+f(y) = \|x\|+\|y\|$.

In fact this is almost never happens.

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No, we cannot, because that map is not a linear map.

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