The Hahn-Banach Theorem allows for the extension of linear maps defined on subspaces of normed spaces to the whole space in a way that respects sublinear bounds. The simplest case is the extension of bounded linear functionals from subspaces to the whole space. We prove this result here, first for real spaces and then for complex spaces. The proof requires use of Zorn’s Lemma, unless we assume that the underlying space is separable.
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