1. You must describe the mapp which has the fixed point - at least where this mapp is defined.
2. The topological and /or the metric properties of this mapp.
If you clear this particulars, you can receive an useful answer.
In Example 2 we can give the interpretation E= R^2, z=(x,y), ||z_1,z_2|| is 2 multiply of the area of triangle z_10z_2, where 0 is origin.
PS. If a triangle is specified by vectors u and v originating at one vertex, then the area is given by half that of the corresponding parallelogram, i.e.,
A = 1/2|det(uv)|= 1/2|uxv|,
where det(A) is the determinant and uxv is a two-dimensional cross product.
This 2-norm corresponds to a metric on the (or a) projective space on Rn - {0}: this is the quotient space for the equivalence relation x equiv y if and only if x = a.y with a non-zero. As Andrey Zahariev notes, fixed points are related to functions. But how are these functions supposed to relate to the 2-norm? Is there a continuity requirement? Or a 2-norm contraction property? Or some other structure?
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