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<title>Spherically complete field</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Spherically complete field</span></span>
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<p>In mathematics, a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i> with an <a href="Absolute_value#Fields" title="Absolute value">absolute value</a> is called <b>spherically complete</b> if the <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of every <a href="Decreasing_sequence" class="mw-redirect" title="Decreasing sequence">decreasing sequence</a> of <a href="Ball_(mathematics)" title="Ball (mathematics)">balls</a> (in the sense of the metric induced by the absolute value) is nonempty:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\in {\mathbf {N} }}B_{n}\neq \emptyset .}">
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<annotation encoding="application/x-tex">{\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\in {\mathbf {N} }}B_{n}\neq \emptyset .}</annotation>
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</math></span><img src="./016690cc8c3733f54b4ad32cf44f25faa78dac6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.009ex; height:5.676ex;" alt="{\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\in {\mathbf {N} }}B_{n}\neq \emptyset .}" loading="lazy"></span></dd></dl>
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</p><p>The definition can be adapted also to a field <i>K</i> with a <a href="Valuation_(algebra)" title="Valuation (algebra)">valuation</a> <i>v</i> taking values in an arbitrary ordered abelian group: (<i>K</i>,<i>v</i>) is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection.
</p><p>Spherically complete fields are important in <a href="Archimedean_property" title="Archimedean property">nonarchimedean</a> <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, since many results analogous to theorems of classical functional analysis require the base field to be spherically complete.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<ul><li>Any <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> field is spherically complete. This includes, in particular, the fields <b>Q</b><sub><i>p</i></sub> of <a href="P-adic_number" title="P-adic number">p-adic numbers</a>, and any of their finite extensions.</li>
<li>Every spherically complete field is <a href="Complete_metric_space" title="Complete metric space">complete</a>. On the other hand, <b>C</b><sub><i>p</i></sub>, the <a href="Complete_metric_space" title="Complete metric space">completion</a> of the <a href="Algebraic_closure" title="Algebraic closure">algebraic closure</a> of <b>Q</b><sub><i>p</i></sub>, is not spherically complete.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>
<li>Any field of <a href="Hahn_series" title="Hahn series">Hahn series</a> is spherically complete.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFVan_der_Put1969" class="citation journal cs1">Van der Put, Marius (1969). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=BSMF_1969__97__309_0">"Espaces de Banach non archimédiens"</a>. <i>Bulletin de la Société Mathématique de France</i>. <b>79</b>: <span class="nowrap">309–</span>320. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.1685">10.24033/bsmf.1685</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0037-9484">0037-9484</a>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSchneider2002" class="citation book cs1">Schneider, P. (2002). <i>Nonarchimedean functional analysis</i>. Springer monographs in mathematics. Berlin ; New York: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-42533-5</bdi>.</cite></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRobert2000" class="citation book cs1">Robert, Alain M. (2000-05-31). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=H6sq_x2-DgoC"><i>A Course in p-adic Analysis</i></a>. Springer Science & Business Media. p. 129. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98669-2</bdi>.</cite></span>
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