Let $x$ be in the normed vector space $X$. Then there is a bounded linear functional $f$ on $X$ such that $$f(x) = ||f|| ||x||.$$

Are there other proofs of this proposition, aside from the proof using Hahn-Banach to extend a function on the subspace of multiples of $x$ sending $\lambda x$ to $\lambda ||x||$?


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