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So the projective plane $\mathbb{RP}^2$ is not a vector space. Is it still isomorphic to its dual? If not, is there at least an invertible map that takes $\mathbb{RP}^2$ to its dual?

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My understanding of the term "dual" in projective geometry is that it only makes sense in an ambient projective space. What definition of "dual" are you using here? – Qiaochu Yuan Jul 28 '10 at 1:59
up vote 3 down vote accepted

Yes, though the word "dual" is somewhat questionable. If you mean is $\mathbb{P}(\mathbb{R}^3)$, the standard projective plane, isomorphic to $\mathbb{P}((\mathbb{R}^3)^*)$, the projectivization of the dual, then yes, it follows from the isomorphism of vector spaces.

Much more interestingly, the duality allows you to switch points and lines in theorems, such as the Mystic Hexagon and Brianchon's Theorem.

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Yeah, I'm referring to $P(\mathbb{R}^3)$ and $P((\mathbb{R}^3)^*)$. So how do you define the isomorphism? – Adeel Jul 28 '10 at 2:06
You just take the isomorphism from to that you get by picking a basis, and that gives you an isomorphism (say, as smooth manifolds or real algebraic varieties). To clarify slightly, it induces an isomorphism by just applying it to the homogeneous coordinates. – Charles Siegel Jul 28 '10 at 2:14

This is a comment more than an answer, mainly in response to Qiaochu's question, but I don't have sufficient rep. to comment.

The dual to a projective plane is the set of all lines in the plane, which itself is a projective plane (as hinted at in Charles Siegel's answer). This is an important concept in classical projective geometry. (Concretely, the equation for a line has the form a x + b y + c z = 0, where a,b, and c are some parameters, not all zero, and x,y,z are homogeneous coords. for the points in the proj. plane. The set of all such lines can thus be thought of as the set of all (a,b,c) not all zero; but note that simultaneously multiplying a, b, and c by a non-zero scalar doesn't change the solution set, i.e. doesn't change the line, so the line should really be thought of as corresponding to the homogeneous coordinates (a:b:c); thus the set of all lines is again a projective plane.)

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