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p-adic analysis
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <i>p</i>-adic analysis is a branch of <a href="/facts/Number_theory/zhoa7zrc">number theory</a> that studies functions of <a href="/facts/P-adic_number/SDgDbkLF"><i>p</i>-adic numbers</a>. Along with the more classical fields of <a href="/facts/Real_analysis/12WjlXCW">real</a> and <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, which deal, respectively, with functions on the <a href="/facts/Real_numbers/R02gw5Pb">real</a> and <a href="/facts/Complex_numbers/2w7mqI7e">complex</a> numbers, it belongs to the discipline of <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>. </p><p>The theory of complex-valued numerical functions on the <i>p</i>-adic numbers is part of the theory of <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact groups</a> (<a href="/facts/Abstract_harmonic_analysis/CyG5bIPb">abstract harmonic analysis</a>). The usual meaning taken for <i>p</i>-adic analysis is the theory of <i>p</i>-adic-valued functions on spaces of interest. </p><p>Applications of <i>p</i>-adic analysis have mainly been in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, where it has a significant role in <a href="/facts/Diophantine_geometry/H2x4wzUE">diophantine geometry</a> and <a href="/facts/Diophantine_approximation/WGVMuWF4">diophantine approximation</a>. Some applications have required the development of <i>p</i>-adic <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and <a href="/facts/Spectral_theory/8sAEEIMM">spectral theory</a>. In many ways <i>p</i>-adic analysis is less subtle than <a href="/facts/Classical_analysis/XXeR26Ih">classical analysis</a>, since the <a href="/facts/Ultrametric_inequality/QhJQBG9N">ultrametric inequality</a> means, for example, that convergence of <a href="/facts/Infinite_series/yDnlsgIe">infinite series</a> of <i>p</i>-adic numbers is much simpler. <a href="/facts/Topological_vector_space/XRex1ChN">Topological vector spaces</a> over <i>p</i>-adic fields show distinctive features; for example aspects relating to <a href="/facts/Convex_set/vdAuJRJl">convexity</a> and the <a href="/facts/Hahn%E2%80%93Banach_theorem/3AUHusvn">Hahn–Banach theorem</a> are different. </p>