Point-line duality (or point-function duality)
The dual space H* of a Hilbert space H is the collection of all the linear functions over H. Every point a in H defines a function fa over H as fa(x) = < a, x>. Now let's pick a point b in H such that fa(b) > 0. The new point b defines another function fb that takes positive value at the first point a by definition as fb(a)=< b, a>=fa(b)>0. Thus the area fa(x) > 0 corresponds to the collection of linear functions that takes positive values at a. If we put more points a1, a2, a3,... we can see how the intersection of these areas changes. In this example, I lifted the coordinate of points in three dimension so that it looks symmetric around the origin. The correspondence can be written as follows:
Point: (a,b) <--> Line: ax + by + 1 =0
Keyboard instructions
- P and G: draw Point(P) or Polygon(G)
- 9 and 0: magnify(9) or reduce(0) the scale in the left panel (the other way around in the right)