Ordered topological vector space

<h2 id="normal-cone">Normal cone</h2>
Main article: <a href="/facts/Normal_cone_(functional_analysis)/fYS2JVmb">Normal cone (functional analysis)</a>
If C is a cone in a TVS X then C is normal if 
 
 
 
 
 
 U
 
 
 =
 
 
 [
 
 
 U
 
 
 ]
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {U}}=\left[{\mathcal {U}}\right]_{C}}
 
, where 
 
 
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathcal {U}}}
 
 is the neighborhood filter at the origin, 
 
 
 
 
 
 [
 
 
 U
 
 
 ]
 
 
 C
 
 
 =
 
 {
 
 
 [
 U
 ]
 
 :
 U
 ∈
 
 
 U
 
 
 
 }
 
 
 
 {\displaystyle \left[{\mathcal {U}}\right]_{C}=\left\{\left[U\right]:U\in {\mathcal {U}}\right\}}
 
, and 
 
 
 
 [
 U
 
 ]
 
 C
 
 
 :=
 
 (
 
 U
 +
 C
 
 )
 
 ∩
 
 (
 
 U
 −
 C
 
 )
 
 
 
 {\displaystyle [U]_{C}:=\left(U+C\right)\cap \left(U-C\right)}
 
 is the <a href="/facts/Cone-saturated/sIaDXRLu">C-saturated</a> hull of a subset U of X.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
If C is a cone in a TVS X (over the real or complex numbers), then the following are equivalent:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 

<ol><li>C is a normal cone.</li>
<li>For every filter 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 in X, if 
 
 
 
 lim
 
 
 F
 
 
 =
 0
 
 
 {\displaystyle \lim {\mathcal {F}}=0}
 
 then 
 
 
 
 lim
 
 
 [
 
 
 F
 
 
 ]
 
 
 C
 
 
 =
 0
 
 
 {\displaystyle \lim \left[{\mathcal {F}}\right]_{C}=0}
 
.</li>
<li>There exists a neighborhood base 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 in X such that 
 
 
 
 B
 ∈
 
 
 B
 
 
 
 
 {\displaystyle B\in {\mathcal {B}}}
 
 implies 
 
 
 
 
 
 [
 
 B
 ∩
 C
 
 ]
 
 
 C
 
 
 ⊆
 B
 
 
 {\displaystyle \left[B\cap C\right]_{C}\subseteq B}
 
.</li></ol>
and if X is a vector space over the reals then also:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<ol><li>There exists a neighborhood base at the origin consisting of convex, <a href="/facts/Balanced_set/Y2ar7E6g">balanced</a>, <a href="/facts/Cone-saturated/sIaDXRLu">C-saturated</a> sets.</li>
<li>There exists a generating family 
 
 
 
 
 
 P
 
 
 
 
 {\displaystyle {\mathcal {P}}}
 
 of semi-norms on X such that 
 
 
 
 p
 (
 x
 )
 ≤
 p
 (
 x
 +
 y
 )
 
 
 {\displaystyle p(x)\leq p(x+y)}
 
 for all 
 
 
 
 x
 ,
 y
 ∈
 C
 
 
 {\displaystyle x,y\in C}
 
 and 
 
 
 
 p
 ∈
 
 
 P
 
 
 
 
 {\displaystyle p\in {\mathcal {P}}}
 
.</li></ol>
If the topology on X is locally convex then the closure of a normal cone is a normal cone.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h3>Properties</h3>
If C is a normal cone in X and B is a bounded subset of X then 
 
 
 
 
 
 [
 B
 ]
 
 
 C
 
 
 
 
 {\displaystyle \left[B\right]_{C}}
 
 is bounded; in particular, every interval 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 is bounded.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 
If X is Hausdorff then every normal cone in X is a proper cone.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="properties">Properties</h2>
<ul><li>Let X be an <a href="/facts/Ordered_vector_space/yN9VfKMa">ordered vector space</a> over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>Let X be an ordered vector space over the reals with positive cone C. Then the following are equivalent:<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li></ul>
<ol><li>the order of X is regular.</li>
<li>C is sequentially closed for some Hausdorff locally convex TVS topology on X and 
 
 
 
 
 X
 
 +
 
 
 
 
 {\displaystyle X^{+}}
 
 distinguishes points in X</li>
<li>the order of X is Archimedean and C is normal for some Hausdorff locally convex TVS topology on X.</li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Generalised_metric/0yyMaqW6">Generalised metric</a> – Metric geometry</li>
<li><a href="/facts/Order_topology_(functional_analysis)/pL01DEh1">Order topology (functional analysis)</a> – Topology of an ordered vector space</li>
<li><a href="/facts/Ordered_field/an0prSgG">Ordered field</a> – Algebraic object with an ordered structure</li>
<li><a href="/facts/Ordered_group/xoLNpEqH">Ordered group</a> – Group with a compatible partial orderPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Ordered_ring/4dGJVBIK">Ordered ring</a> – ring with a compatible total orderPages displaying wikidata descriptions as a fallback</li>
<li><a href="/facts/Ordered_vector_space/yN9VfKMa">Ordered vector space</a> – Vector space with a partial order</li>
<li><a href="/facts/Partially_ordered_space/06RwXYXo">Partially ordered space</a> – Partially ordered topological space</li>
<li><a href="/facts/Riesz_space/FX7Mwfxe">Riesz space</a> – Partially ordered vector space, ordered as a lattice</li>
<li><a href="/facts/Topological_vector_lattice/q5BLo3cr">Topological vector lattice</a></li></ul>

<ul><li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Schaefer & Wolff 1999, pp. 222–225. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Schaefer & Wolff 1999, pp. 222–225. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Schaefer & Wolff 1999, pp. 222–225. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Ordered topological vector space open-in-new

Ordered topological vector space