Talk:Polar set

Polar set is not defined

Currently, the article does not say what a polar set is. It just says that a polar set is a set in the dual space if it is a polar set of a subset of a vector space. -- Kjkolb 01:09, 7 May 2007 (UTC)[reply]

I tried to clarify the article. MathMartin 16:57, 8 May 2007 (UTC)[reply]

Theorem of bipolars

I suggest adding something like the following

``* By the theorem of bipolars $A^{00}$ is convex weak-closure of $A\cup \{0\}$ , that is the smallest weak-closed convex set containing $A$ and 0.

This asks for right links to the definition of weak topology, sometimes denoted as $\sigma (X,Y)$ Matumba (talk) 01:27, 27 April 2008 (UTC)[reply]

As it stands, the article statement is wrong: $A^{00}$ is not equal to the absolutely convex envelope (take the open unit ball in the reals for example), but to the weak-closure of it. Chrystomath2 (talk) 09:03, 15 April 2016 (UTC)[reply]

Confusion

The page says "There are at least three competing definitions of the polar of a set.", lists two of them, and then talk about properties. Does these properties hold for all definitions or what is intended?

The page says "Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.", and then makes this definition in the last property: "For a closed convex cone C, the dual cone is the polar of C". — Preceding unsigned comment added by 5.57.55.92 (talk) 10:34, 24 January 2020 (UTC)[reply]

And I would add, it is not very smart, and quite confusing, adopting the same name and notation for the polar of the polar (i.e. the bi-polar) of $A\subset X$ and the pre-polar of the polar. Polar sends to the dual, pre-polar sends backwards to the pre-dual. The polar of a subset of $B\subset X^{*}$ is a subset of $X^{**}$ ; the pre-polar is just the trace of its polar on $X$ , via the isometric embedding $X\to X^{**}$ . Also, the weak topology of $X$ ( $\sigma (X,X^{*})$ ) is just the trace on $X$ of the weak* topology of $X^{**}$ ( $\sigma (X^{**},X^{*})$ ). pm a 23:41, 9 November 2022 (UTC)[reply]