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title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">January 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For non-archimedean analytic spaces, see <a href="/wiki/Berkovich_space" title="Berkovich space">Berkovich space</a>.</div> <p>An <b>analytic space</b> is a generalization of an <a href="/wiki/Analytic_manifold" title="Analytic manifold">analytic manifold</a> that allows <a href="/wiki/Singularity_(mathematics)" title="Singularity (mathematics)">singularities</a>. An analytic space is a space that is <a href="/wiki/Local_property" title="Local property">locally</a> the same as an <a href="/wiki/Analytic_variety" class="mw-redirect" title="Analytic variety">analytic variety</a>. They are prominent in the study of <a href="/wiki/Several_complex_variables" class="mw-redirect" title="Several complex variables">several complex variables</a>, but they also appear in other contexts. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fix a field <i>k</i> with a valuation. Assume that the field is complete and not discrete with respect to this valuation. For example, this includes <b>R</b> and <b>C</b> with respect to their usual absolute values, as well as fields of <a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a> with respect to their natural valuations. </p><p>Let <i>U</i> be an open subset of <i>k</i><sup><i>n</i></sup>, and let <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>k</i></sub> be a collection of <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> on <i>U</i>. Denote by <i>Z</i> the common vanishing locus of <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>k</i></sub>, that is, let <i>Z</i> = { <i>x</i> | <i>f</i><sub>1</sub>(<i>x</i>) = ... = <i>f</i><sub><i>k</i></sub>(<i>x</i>) = 0 }. <i>Z</i> is an analytic variety. </p><p>Suppose that the structure <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of <i>U</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{U}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a9cef2217740befda1b31bb9a65ec7cb8a1d8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{U}}"></span>. Then <i>Z</i> has a structure sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{Z}={\mathcal {O}}_{U}/{\mathcal {I}}_{Z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{Z}={\mathcal {O}}_{U}/{\mathcal {I}}_{Z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c744e65a2066269d39e3e3032ccbe5cc17330465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.561ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{Z}={\mathcal {O}}_{U}/{\mathcal {I}}_{Z}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}_{Z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}_{Z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d6ba7561b91b5d1fcf6f973d5094cfe641953f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.069ex; width:2.756ex; height:2.509ex;" alt="{\displaystyle {\mathcal {I}}_{Z}}"></span> is the ideal generated by <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>k</i></sub>. In other words, the structure sheaf of <i>Z</i> consists of all functions on <i>U</i> modulo the possible ways they can differ outside of <i>Z</i>. </p><p>An <b>analytic space</b> is a <a href="/wiki/Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {O}}_{X})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46716c1fec068ffe9981107a5a215fa1fc9e7d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.306ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})}"></span> such that around every point <i>x</i> of <i>X</i>, there exists an open neighborhood <i>U</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,{\mathcal {O}}_{U})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>U</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (U,{\mathcal {O}}_{U})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1a04c47b4e62eb4d313c553b27d3dc9b86f196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.969ex; height:2.843ex;" alt="{\displaystyle (U,{\mathcal {O}}_{U})}"></span> is isomorphic (as locally ringed spaces) to an analytic variety with its structure sheaf. Such an isomorphism is called a <b>local model</b> for <i>X</i> at <i>x</i>. </p><p>An <b>analytic mapping</b> or <b>morphism</b> of analytic spaces is a morphism of locally ringed spaces. </p><p>This definition is similar to the definition of a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>. The only difference is that for a scheme, the local models are <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectra of rings</a>, whereas for an analytic space, the local models are analytic varieties. Because of this, the basic theories of analytic spaces and of schemes are very similar. Furthermore, analytic varieties have much simpler behavior than arbitrary commutative rings (for example, analytic varieties are defined over fields and are always finite-dimensional), so analytic spaces behave very similarly to finite-type schemes over a field. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_results">Basic results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=2" title="Edit section: Basic results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every point in an analytic space has a local dimension. The dimension at <i>x</i> is found by choosing a local model at <i>x</i> and determining the local dimension of the analytic variety at the point corresponding to <i>x</i>. </p><p>Every point in an analytic space has a <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a>. If <i>x</i> is a point of <i>X</i> and <i>m<sub>x</sub></i> is ideal sheaf of all functions vanishing at <i>x</i>, then the cotangent space at <i>x</i> is <span class="nowrap"><i>m<sub>x</sub></i> / <i>m<sub>x</sub></i><sup>2</sup></span>. The tangent space is <span class="nowrap">(<i>m<sub>x</sub></i> / <i>m<sub>x</sub></i><sup>2</sup>)<sup>*</sup></span>, the dual vector space to the cotangent space. Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces. </p><p>The dimension of the tangent space at <i>x</i> is called the <b><a href="/wiki/Embedding_dimension" class="mw-redirect" title="Embedding dimension">embedding dimension</a></b> at <i>x</i>. By looking at a local model it is easy to see that the dimension is always less than or equal to the embedding dimension. </p> <div class="mw-heading mw-heading2"><h2 id="Smoothness">Smoothness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=3" title="Edit section: Smoothness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An analytic space is called <b>smooth</b> at <i>x</i> if it has a local model at <i>x</i> which is an open subset of <i>k</i><sup><i>n</i></sup> for some <i>n</i>. The analytic space is called smooth if it is smooth at every point, and in this case it is an <a href="/wiki/Analytic_manifold" title="Analytic manifold">analytic manifold</a>. The subset of points at which an analytic space is not smooth is a closed analytic subset. </p><p>An analytic space is <b>reduced</b> if every local model for the space is defined by a radical sheaf of ideals. An analytic space <i>X</i> which isn't reduced has a <b>reduction</b> <i>X</i><sub>red</sub>, a reduced analytic space with the same underlying topological space. There is a canonical morphism <span class="nowrap"><i>r</i> : <i>X</i><sub>red</sub> → <i>X</i></span>. Every morphism from <i>X</i> to a reduced analytic space factors through <i>r</i>. </p><p>An analytic space is <b>normal</b> if every stalk of the structure sheaf is a normal ring (meaning an integrally closed integral domain). In a normal analytic space, the singular locus has codimension at least two. When <i>X</i> is a local complete intersection at <i>x</i>, then <i>X</i> is normal at <i>x</i>. </p><p>Non-normal analytic spaces can be smoothed out into normal spaces in a canonical way. This construction is called the <b>normalization</b>. The normalization <i>N</i>(<i>X</i>) of an analytic space <i>X</i> comes with a canonical map <span class="nowrap">ν : <i>N</i>(<i>X</i>) → <i>X</i></span>. Every dominant morphism from a normal analytic space to <i>X</i> factors through ν. </p> <div class="mw-heading mw-heading2"><h2 id="Coherent_sheaves">Coherent sheaves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=4" title="Edit section: Coherent sheaves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An analytic space is <b>coherent</b> if its structure sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}"></span> is a <a href="/wiki/Coherent_sheaf" title="Coherent sheaf">coherent sheaf</a>. A coherent sheaf of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}"></span>-modules is called a <b>coherent analytic sheaf</b>. For example, on a coherent space, locally free sheaves and sheaves of ideals are coherent analytic sheaves. </p><p>Analytic spaces over algebraically closed fields are coherent. In the complex case, this is known as the <a href="/wiki/Oka_coherence_theorem" title="Oka coherence theorem">Oka coherence theorem</a>. This is not true over non-algebraically closed fields; there are examples of real analytic spaces that are not coherent. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=5" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In some situations, the concept of an analytic space is too restrictive. This is often because the ground field has additional structure that is not captured by analytic sets. In these situations, there are generalizations of analytic spaces which allow more flexibility in the local model spaces. </p><p>For example, over the real numbers, consider the circle <span class="nowrap"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 1</span>. The circle is an analytic subset of the analytic space <b>R</b><sup>2</sup>. But its projection onto the <i>x</i>-axis is the closed interval <span class="nowrap">[−1, 1]</span>, which is not an analytic set. Therefore the image of an analytic set under an analytic map is not necessarily an analytic set. This can be avoided by working with <a href="/wiki/Subanalytic_set" title="Subanalytic set">subanalytic sets</a>, which are much less rigid than analytic sets but which are not defined over arbitrary fields. The corresponding generalization of an analytic space is a subanalytic space. (However, under mild <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a> hypotheses, it turns out that subanalytic spaces are essentially equivalent to subanalytic sets.) </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Analytic_variety" class="mw-redirect" title="Analytic variety">Analytic variety</a></li> <li><a href="/wiki/Complex_analytic_space" class="mw-redirect" title="Complex analytic space">Complex analytic space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analytic_space&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFOnishchik2001" class="citation cs2">Onishchik, A. 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(2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Real-analytic_space">"Real-analytic space"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Real-analytic+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Ponomarev&rft.aufirst=D.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DReal-analytic_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAnalytic+space" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Analytic_space&oldid=1215194898">https://en.wikipedia.org/w/index.php?title=Analytic_space&oldid=1215194898</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Algebraic_geometry" title="Category:Algebraic geometry">Algebraic geometry</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_January_2024" title="Category:Articles lacking in-text citations from January 2024">Articles lacking in-text citations from January 2024</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 23 March 2024, at 18:08<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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