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| In [[mathematics]], the '''Banach–Mazur theorem''' is a [[theorem]] of [[functional analysis]]. Very roughly, it states that most [[well-behaved]] [[normed spaces]] are [[Linear subspace|subspace]]s of the space of [[continuous function (topology)|continuous]] [[Path (topology)|paths]]. It is named after [[Stefan Banach]] and [[Stanisław Mazur]].
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| ==Statement of the theorem==
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| Every [[real number|real]], [[separable space|separable]] [[Banach space]] (''X'', || ||) is [[Isometry|isometrically isomorphic]] to a [[Closed set|closed]] subspace of ''C''<sup> 0</sup>([0, 1]; '''R'''), the space of all [[continuous function]]s from the unit [[Interval (mathematics)|interval]] into the real line.
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| ==Comments==
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| On the one hand, the Banach–Mazur theorem seems to tell us that the seemingly vast collection of all separable Banach spaces is not that vast or difficult to work with, since a separable Banach space is "just" a collection of continuous paths. On the other hand, the theorem tells us that ''C''<sup> 0</sup>([0, 1]; '''R''') is a "really big" space, big enough to contain every possible separable Banach space.
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| Non-separable Banach spaces cannot embed isometrically in the separable space ''C''([0, 1]), but for every Banach space ''X'', one can find a [[compact space|compact]] [[Hausdorff space]] ''K'' and an isometric linear embedding ''j'' of ''X'' into the space ''C''(''K'') of scalar continuous functions on ''K''. The simplest choice is to let ''K'' be the [[Unit sphere|unit ball]] of the [[Dual space#Continuous dual space|continuous dual]] ''X''', equipped with the [[Weak topology|w*-topology]]. This unit ball ''K'' is then compact by the [[Banach-Alaoglu theorem|Banach–Alaoglu theorem]]. The embedding ''j'' is introduced by saying that for every {{nowrap|''x'' ∈ ''X''}}, the continuous function {{nowrap|''j''(''x'')}} on ''K'' is defined by
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| :<math> \forall x' \in K, \ \ j(x)(x') = x'(x).</math>
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| The mapping ''j'' is linear, and it is isometric by the [[Hahn-Banach theorem|Hahn–Banach theorem]].
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| ==Stronger versions of the theorem==
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| Let's write ''C''<sup> k</sup>[0, 1] for ''C''<sup> k</sup>([0, 1]; '''R'''). In 1995, Luis Rodríguez-Piazza proved that the isometry ''i'' : ''X'' → ''C''<sup> 0</sup>[0, 1] can be chosen so that every non-zero function in the [[image (mathematics)|image]] ''i''(''X'') is [[nowhere differentiable]]. Put another way, if ''D'' denotes the subset of ''C''<sup> 0</sup>[0, 1] consisting of those functions that are differentiable at least one point of [0, 1], then ''i'' can be chosen so that ''i''(''X'') ∩ ''D'' = {0}. This conclusion applies to the space ''C''<sup> 0</sup>[0, 1] itself, hence there exists a [[linear map]] ''i'' from ''C''<sup> 0</sup>[0, 1] to itself that is an isometry onto its image, such that image under ''i'' of ''C''<sup> 1</sup>[0, 1] (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects ''D'' only at 0: thus the space of smooth functions (w.r. to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the (metrically incomplete) space of smooth functions is dense in ''C''<sup> 0</sup>[0, 1].
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| ==References==
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| * {{cite book
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| | author=Bessaga, Czesław, & Pełczyński, Aleksander
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| | title=Selected topics in infinite-dimensional topology
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| | location=Warszawa
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| | publisher=PWN
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| | year=1975
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| }}
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| * {{cite journal
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| | last = Rodríguez-Piazza
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| | first = Luis
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| | title = Every separable Banach space is isometric to a space of continuous nowhere differentiable functions
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| | journal = [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]]
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| | volume = 123
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| | year = 1995
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| | pages = 3649–3654
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| | doi = 10.2307/2161889
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| | issue = 12
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| |publisher = [[American Mathematical Society]]
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| | jstor = 2161889
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| }}
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Banach-Mazur Theorem}}
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| [[Category:Functional analysis]]
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| [[Category:Continuous mappings]]
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| [[Category:Theorems in functional analysis]]
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