I'd like to use the following fundamental result (1) in my paper but I don't know a good reference. Could anyone help?

Setup: for a set $A$ in a vector space $V$, its algebraic closure $A^{ac}$ consists of points $x\in V$ that there is some $v\in V$ so that for any $\epsilon>0$, there is some $t\in [0,\epsilon]$ such that $x+tv\in A$. I believe the following result (1) is true and fundamental:

(1) If $V$ is finite-dimensional and $A$ is convex, then $A^{ac}$ is the same as the closure of $A$ in the usual norm topology.

The references (books) on convex analysis in English that I know do not talk about algebraic closure since they either focus on finite-dimensional spaces or normed vector spaces. Does anyone know a suitable reference?

Alternatively, a reference to any of the following results (2)(3) would be good enough. This involves the notion of an algebraically closed set, which is a set $A$ such that for any $x\notin A$ and any $v\in V$, there is some $\epsilon>0$ so that $x+tv \notin A$ for all $t\in [0,\epsilon]$.

(2) If $V$ is finite-dimensional, then a convex algebraically closed set $A$ is closed in the usual norm topology.

(3) For an algebraically closed convex set $A$, any point $x$ outside of $A$ is strongly separated from $A$, that is, there is a linear functional $f$ and a number $c$ so that $f(x)<c\le f(a)$ for all $a\in A$.

It is easy to note that (2) implies (1) since in a finite-dimensional space, the algebraic closure of a convex set is algebraically closed, thus $A^{ac}$ is closed and contains the closure $\bar{A}$; conversely $A^{ac}\subset (\bar{A})^{ac}=\bar{A}$.

(2) follows from (3) since the strong separation provides each point $x\notin A$ an open neighborhood (an open half space) disjoint from $A$.


1 Answer 1


I have found a reference myself:

Richard B. Holmes, Geometric Functional Analysis and its Applications, (GTM 24). Page 9 and Page 59.

In the book, a related notion $\mathrm{lin}(A)$ is defined, which is the union of $A$ and points linearly accessible from $A$. Here a point $x$ is linearly accessible from $A$ if there is some $a\in A$ such that $[a,x)\subset A$. It is easy to note that for any $A$ convex, the algebraic closure defined in the question coincides with $\mathrm{lin}(A)$.

It is commented in the book on Page 9 part D that $\mathrm{lin}(A)$ is the (topological) closure of $A$ in a finite-dimensional space if $A$ is convex. A more general statement is part (c) of the Lemma on Page 59, which states

$\mathrm{lin}(A)=\bar{A}$ for any convex set $A$ with nonempty interior in a topological vector space.

If $A$ lies in a finite-dimensional space, then the statement above applies to the affine hull of $A$ (i.e. the smallest affine subspace containing $A$), implying statement (1) in the question.


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