I would like to know more about the geometry of $\mathbb{R}^2$ equipped with the following inner product $(\mathbf{v},\mathbf{u})=\|\mathbf v\|\cdot \|\mathbf u\|\cos(2\alpha)$, where $\alpha$ is the angle between the vectors $\mathbf v$ and $\mathbf u$. This is not a true inner product since $(a\mathbf v,\mathbf u)=|a|(\mathbf v,\mathbf u)$. In a way, this is an inner product on the space of directions in $\mathbb{R}^2$. Has this been studied, and where could I start looking into literature? I am particularly interested in the possibility of representing the space in terms of spinors.
Pseudo inner product question
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inner-product-space
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0Yeah, your inner product is really curious, I found this somewhat related article on standard inner product and angles http://www.springerlink.com/content/472w17185p0771w0/ – 2012-04-11
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0This looks like a norm on $\mathbb R^2$ that is not induced by an inner product (I haven't checked if this satisfies the triangle inequality). If so, then it induces the usual topology on $\mathbb R^2$, because all norms on finite-dimensional spaces are equivalent. – 2012-08-21