Tagged Questions
0
votes
0answers
28 views
Is this question stated wrong?
I'm trying to check whether this question might be worded wrong, and here it is:
Show that if $A$ is a convex subset of a topological vector space $X$, $u \in A^o$ (the interior of $A$), $v \in ...
1
vote
1answer
43 views
Questions regarding internal and interior points for a convex subset of a topological vector space
Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
0
votes
0answers
61 views
Is a closed set in a TVS over $\mathbb{R}$ convex?
From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The ...
1
vote
0answers
39 views
Two different opinions on whether a topological vector space is a uniform space
Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex:
3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the
standard convexity and ...
2
votes
3answers
111 views
Balanced but not convex?
In a topological vector space $X$, a subset $S$ is convex if \begin{equation}tS+(1-t)S\subset S\end{equation} for all $t\in (0,1)$.
$S$ is balanced if \begin{equation}\alpha S\subset S\end{equation} ...
0
votes
0answers
85 views
Is the dual cone of the dual cone equal to the original cone? [duplicate]
Possible Duplicate:
Dual of a dual cone
I try to prove the following statement:
Let $V$ be a finite-dimensional ordered topological vector space ($V^{**} \cong V$) with a closed positive ...
5
votes
1answer
558 views
Convex functions and families of affine functions
I know that the supremum of a family of affine functions is convex. Just wondering if it is true (and if so how one proves) that the converse -- any $C^1$ convex function is the supremum of some ...
7
votes
2answers
700 views
Dual of a dual cone
Any hint on how to prove the following please:
Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$.
Thanks!
