Polar set

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. (where this is equivalent to s A = A for all unit length scalar s). In particular, all definitions of the polar of A agree when A is a balanced set (which is often, but not always, the case) so that often, which of these competing definitions is used is immaterial. However, these differences in the definitions of the "polar" of a set A do sometimes introduce subtle or important technical differences when A is not necessarily balanced. Specialization for the canonical duality Algebraic dual space If X is any vector space then let X^{\#} denote the algebraic dual space of X, which is the set of all linear functionals on X. The vector space X^{\#} is always a closed subset of the space \mathbb{K}^X of all \mathbb{K}-valued functions on X under the topology of pointwise convergence so when X^{\#} is endowed with the subspace topology, then X^{\#} becomes a Hausdorff complete locally convex topological vector space (TVS). For any subset A \subseteq X, let \begin{alignat}{4} A^{\#} := A^{\circ, \#} :=& \left\{f \in X^{\#} ~:~ \sup_{a \in A} |f(a)| \leq 1\right\} && \\[0.7ex] =& \left\{f \in X^{\#} ~:~ \sup |f(A)| \leq 1\right\} ~~~~&& \text{ where } |f(A)| := \{|f(a)| : a \in A\} \\[0.7ex] =& \left\{f \in X^{\#} ~:~ f(A) \subseteq B_1\right\} ~~~&& \text{ where } B_1 := \{s \in \mathbb{K} : |s| \leq 1\}.\\[0.7ex] \end{alignat} If A \subseteq B \subseteq X are any subsets then B^{\#} \subseteq A^{\#} and A^{\#} = [\operatorname{cobal} A]^{\#}, where \operatorname{cobal} A denotes the convex balanced hull of A. For any finite-dimensional vector subspace Y of X, let \tau_Y denote the Euclidean topology on Y, which is the unique topology that makes Y into a Hausdorff topological vector space (TVS). If A_{\cup \operatorname{cl} \operatorname{Finite}} denotes the union of all closures \operatorname{cl}_{\left(Y, \tau_Y\right)} (Y \cap A) as Y varies over all finite dimensional vector subspaces of X, then A^{\#} = \left[A_{\cup \operatorname{cl} \operatorname{Finite}}\right]^{\#} (see this footnote for an explanation). If A is an absorbing subset of X then by the Banach–Alaoglu theorem, A^{\#} is a weak-* compact subset of X^{\#}. If A \subseteq X is any non-empty subset of a vector space X and if Y is any vector space of linear functionals on X (that is, a vector subspace of the algebraic dual space of X) then the real-valued map :|\,\cdot\,|_A \;:\, Y \,\to\, \Reals defined by \left|x^{\prime}\right|_A ~:=~ \sup \left|x^{\prime}(A)\right| ~:=~ \sup_{a \in A} \left|x^{\prime}(a)\right| is a seminorm on Y. If A = \varnothing then by definition of the supremum, \, \sup \left| x^{\prime}(A) \right| = -\infty \, so that the map \, |\,\cdot\,|_{\varnothing} = -\infty \, defined above would not be real-valued and consequently, it would not be a seminorm. Continuous dual space Suppose that X is a topological vector space (TVS) with continuous dual space X^{\prime}. The important special case where Y := X^{\prime} and the brackets represent the canonical map: \left\langle x, x^{\prime} \right\rangle := x^{\prime}(x) is now considered. The triple \left\langle X, X^{\prime} \right\rangle is the called the associated with X. The polar of a subset A \subseteq X with respect to this canonical pairing is: \begin{alignat}{4} A^{\circ} :=& \left\{x^{\prime} \in X^{\prime} ~:~ \sup_{a \in A} \left|x^{\prime}(a)\right| \leq 1\right\} ~~~~&& \text{ because } \left\langle a, x^{\prime} \right\rangle := x^{\prime}(a) \\[0.7ex] =& \left\{x^{\prime} \in X^{\prime} ~:~ \sup \left|x^{\prime}(A)\right| \leq 1\right\} ~~~~&& \text{ where } \left|x^{\prime}(A)\right| := \left\{\left|x^{\prime}(a)\right| : a \in A\right\} \\[0.7ex] =& \left\{x^{\prime} \in X^{\prime} ~:~ x^{\prime}(A) \subseteq B_1\right\} ~~~~&& \text{ where } B_1 := \{s \in \mathbb{K} : |s| \leq 1\}.\\[0.7ex] \end{alignat} For any subset A \subseteq X, A^{\circ} = \left[\operatorname{cl}_X A\right]^{\circ} where \operatorname{cl}_X A denotes the closure of A in X. The Banach–Alaoglu theorem states that if A \subseteq X is a neighborhood of the origin in X then A^{\circ} = A^{\#} and this polar set is a compact subset of the continuous dual space X^{\prime} when X^{\prime} is endowed with the weak-* topology (also known as the topology of pointwise convergence). If A satisfies s A \subseteq A for all scalars s of unit length then one may replace the absolute value signs by \operatorname{Re} (the real part operator) so that: \begin{alignat}{4} A^{\circ} = A^r :=& \left\{x^{\prime} \in X^{\prime} ~:~ \sup_{a \in A} \operatorname{Re} x^{\prime}(a) \leq 1\right\} \\[0.7ex] =& \left\{x^{\prime} \in X^{\prime} ~:~ \sup \operatorname{Re} x^{\prime}(A) \leq 1\right\}. \\[0.7ex] \end{alignat} The prepolar of a subset B of Y = X^{\prime} is: {}^{\circ} B := \left\{x \in X ~:~ \sup_{b^{\prime} \in B} \left|b^{\prime}(x)\right| \leq 1\right\} = \{x \in X : \sup |B(x)| \leq 1\} If B satisfies s B \subseteq B for all scalars s of unit length then one may replace the absolute value signs with \operatorname{Re} so that: {}^{\circ} B = \left\{x \in X ~:~ \sup_{b^{\prime} \in B} \operatorname{Re} b^{\prime}(x) \leq 1\right\} = \{x \in X ~:~ \sup \operatorname{Re} B(x) \leq 1\} where B(x) := \left\{b^{\prime}(x) ~:~ b^{\prime} \in B\right\}. The bipolar theorem characterizes the bipolar of a subset of a topological vector space. If X is a normed space and S is the open or closed unit ball in X (or even any subset of the closed unit ball that contains the open unit ball) then S^{\circ} is the closed unit ball in the continuous dual space X^{\prime} when X^{\prime} is endowed with its canonical dual norm. Geometric definition for cones The polar cone of a convex cone A \subseteq X is the set A^{\circ} := \left\{y \in Y ~:~ \sup_{x \in A} \langle x, y \rangle \leq 0\right\} This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point x \in X is the locus \{y ~:~ \langle y, x \rangle = 0\}; the dual relationship for a hyperplane yields that hyperplane's polar point. Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article. ==Properties==

Unless stated otherwise, \langle X, Y \rangle will be a pairing. The topology \sigma(Y, X) is the weak-* topology on Y while \sigma(X, Y) is the weak topology on X. For any set A, A^r denotes the real polar of A and A^{\circ} denotes the absolute polar of A. The term "polar" will refer to the polar. • The (absolute) polar of a set is convex and balanced. • The real polar A^r of a subset A of X is convex but necessarily balanced; A^r will be balanced if A is balanced. • If s A \subseteq A for all scalars s of unit length then A^{\circ} = A^r. • A^{\circ} is closed in Y under the weak-*-topology on Y. • If \mathcal{B} is a base at the origin for a TVS X then X^{\prime} = \bigcup_{B \in \mathbb{B}} \left(B^{\circ}\right). • If X is a locally convex TVS then the polars (taken with respect to \left\langle X, X^{\prime} \right\rangle) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of X^{\prime} (i.e. given any bounded subset H of X^{\prime}_{\sigma}, there exists a neighborhood S of the origin in X such that H \subseteq S^{\circ}). • Conversely, if X is a locally convex TVS then the polars (taken with respect to \langle X, X^{\#} \rangle) of any fundamental family of equicontinuous subsets of X^{\prime} form a neighborhood base of the origin in X. • Let X be a TVS with a topology \tau. Then \tau is a locally convex TVS topology if and only if \tau is the topology of uniform convergence on the equicontinuous subsets of X^{\prime}. The last two results explain why equicontinuous subsets of the continuous dual space play such a prominent role in the modern theory of functional analysis: because equicontinuous subsets encapsulate all information about the locally convex space X's original topology. Set relations • X^{\circ} = X^ = X^r = \{0\} and \varnothing^{\circ} = \varnothing^ = \varnothing^r = Y. • For all scalars s \neq 0, (s A)^{\circ} = \tfrac{1}{s} \left(A^{\circ}\right) and for all real t \neq 0, (t A)^ = \tfrac{1}{t} \left(A^\right) and (t A)^r = \tfrac{1}{t} \left(A^r\right). • A^{\circ\circ\circ} = A^{\circ}. However, for the real polar we have A^{r r r} \subseteq A^r. • For any finite collection of sets A_1, \ldots, A_n, \left(A_1 \cap \cdots \cap A_n\right)^{\circ} = \left(A_1^{\circ}\right) \cup \cdots \cup \left(A_n^{\circ}\right). • If A \subseteq B then B^{\circ} \subseteq A^{\circ}, B^r \subseteq A^r, and B^ \subseteq A^. • An immediate corollary is that \bigcup_{i \in I} \left(A_i^{\circ}\right) \subseteq \left(\bigcap_{i \in I} A_i\right)^{\circ}; equality necessarily holds when I is finite and may fail to hold if I is infinite. • \bigcap_{i \in I} \left(A_i^{\circ}\right) = \left(\bigcup_{i \in I} A_i\right)^{\circ} and \bigcap_{i \in I} \left(A_i^r\right) = \left(\bigcup_{i \in I} A_i\right)^r. • If C is a cone in X then C^{\circ} = \left\{ y \in Y : \langle c, y \rangle = 0 \text{ for all } c \in C \right\}. • If \left(S_i\right)_{i \in I} is a family of \sigma(X, Y)-closed subsets of X containing 0 \in X, then the real polar of \cap_{i \in I} S_i is the closed convex hull of \cup_{i \in I} \left(S_i^r\right). • If 0 \in A \cap B then A^{\circ} \cap B^{\circ} \subseteq 2 \left[(A + B)^{\circ}\right] \subseteq 2\left(A^{\circ} \cap B^{\circ}\right). • For a closed convex cone C in a real vector space X, the polar cone is the polar of C; that is, C^{\circ} = \{y \in Y : \sup_{} \langle C, y \rangle \leq 0\}, where \sup_{} \langle C, y \rangle := \sup_{c \in C} \langle c, y \rangle. ==See also==