The Hausdorff maximal principle states that, in any totally ordered subset is contained in a maximal totally ordered subset.
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The Hausdorff maximal principle is one of many statements equivalent to the axiom of choice over ZF ( Zermelo Fraenkel set theory without the axiom of choice ).
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Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that
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