View source for Algebraic closure (convex analysis)

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{{About||the algebraic closure of a field|Algebraic closure}}
'''Algebraic closure''' of a subset <math>A</math> of a vector space <math>X</math> is the set of all points that are linearly accessible from <math>A</math>. It is denoted by <math>\operatorname{acl}  A</math> or <math>\operatorname{acl}_X  A</math>.

A point <math>x \in X</math> is said to be '''linearly accessible''' from a subset <math>A \subseteq X</math> if there exists some <math>a \in A</math> such that the line segment <math>[a, x) := a + [0, 1) (x-a)</math> is contained in <math>A</math>.

Necessarily, <math>A\subseteq \operatorname{acl}  A \subseteq \operatorname{acl} \operatorname{acl} A \subseteq \overline{A}</math> (the last inclusion holds when ''X'' is equipped by any [[vector topology]], Hausdorff or not).

The set ''A'' is '''algebraically closed''' if <math>A = \operatorname{acl}  A</math>.
The set <math> \operatorname{acl}  A \setminus  \operatorname{aint} A </math> is the '''algebraic boundary''' of ''A'' in ''X''.

==Examples==

The set <math>\Q</math> of [[rational numbers]] is algebraically closed but <math>\Q^c</math> is not algebraically open

If <math>A = \{(x,y) \in \R^2: 0 < y < x^2\} \subseteq \R^2</math> then 
<math>0 \in (\operatorname{acl} \operatorname{acl} A) \setminus \operatorname{acl} A</math>. In particular, the algebraic closure need not be algebraically closed.
Here, <math> \overline{A}=\operatorname{acl} \operatorname{acl} A =  \{(x,y) \in \R^2: 0 \le y \le x^2\} 
   = (\operatorname{acl} A)\cup\{0\}</math>.

However, <math>\operatorname{acl} A =\overline{A}</math> for every finite-dimensional [[convex set]] ''A''.

Moreover, a convex set is algebraically closed if and only if its complement is [[algebraically open]].

== See also ==
* [[Algebraic interior]]
== References ==
{{Reflist}}

==Bibliography==
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->

{{Functional analysis}}
{{Topological vector spaces}}
{{Convex analysis and variational analysis}}

[[Category:Convex analysis]]
[[Category:Functional analysis]]
[[Category:Mathematical analysis]]
[[Category:Topology]]