The polar of a point with respect to a non-degenerate conic is the line containing all points harmonically conjugate to with respect to the points and of intersection of the conic with secants through (cf. Cross ratio). The point is called the pole. If the point lies outside the conic, then the polar passes through the points of contact of the two tangent lines that can be drawn through (see Fig. a). If the point lies on the curve, then the polar is the tangent to the curve at this point. If the polar of the point passes through a point , then the polar of passes through (see Fig. b).
Every non-degenerate conic determines a bijection between the set of points of the projective plane and the set of its straight lines, which is a polarity (a polar transformation). Figures that correspond under this transformation are called mutually polar. A figure coinciding with its polar figure is called autopolar, or self-polar (see, for example, the self-polar triangle in Fig. b).
Analogously one defines the polar (polar plane) of a point with respect to a non-degenerate surface of the second order.
The concept of a polar relative to a conic can be generalized to curves of order . Here, a given point of the plane is put into correspondence with polars with respect to the curve. The first of these polars is a curve of order , the second, which is the polar of the given point relative to the first polar, has order , etc., and, finally, the -st polar is a straight line.
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The polar of a subset in a locally convex topological vector space is the set of functionals in the dual space for which for all (here is the value of at ). The bipolar is the set of vectors in the space for which for all .
The polar is convex, balanced and closed in the weak- topology . The bipolar is the weak closure of the convex balanced hull of the set . In addition, . If is a neighbourhood of zero in the space , then its polar is a compactum in the weak- topology (the Banach–Alaoglu theorem).
The polar of the union of any family of sets in is the intersection of the polars of these sets. The polar of the intersection of weakly-closed convex balanced sets is the closure in the weak- topology of the convex hull of their polars. If is a subspace of , then its polar coincides with the subspace of orthogonal to .
As a fundamental system of neighbourhoods of zero defining the weak- topology of the space one can take the system of sets of the form where runs through all finite subsets of .
A subset of functionals of the space is equicontinuous if and only if it is contained in the polar of some neighbourhood of zero.
|||R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)|
|[a1]||G. Köthe, "Topological vector spaces" , 1 , Springer (1979) (Translated from German)|
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|[a3]||H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)|
Polar. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Polar&oldid=14444