By Jeffrey M. Lemm

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Extra resources for Handbook of the Geometry of Banach Spaces, Volume Volume 2

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Definition A two-dimensional polarity is called hyperbolic or elliptic according to whether it does or does not admit self-conjugate polar elements. The following theorem is instrumental for generalizing the projective notion of a conic [cone of the second degree] so as to include improper (imaginary) elements (see below). 9a A polarity in a bundle induces an involution of conjugate induces an involution of conjugate polar polar points on any line that is not a planes on any line that is not a selfself-conjugate polar line and an involu- conjugate polar line and an involution tion of conjugate polar lines in any of conjugate polar lines in any plane point that is not a self-conjugate polar that is not a self-conjugate polar plane.

I) If α does not pass through S, at least one of the two lines, say g, passes through R. Consider the planes passing through g. 1A). Hence there are infinite many planes through g that contain exactly one second line l of the surface. (ii) If α passes through S, one of the two lines in α, say g, must coincide with RS. Consider the planes passing through g. 1A). Hence there are infinite many planes through g that contain exactly one second line l of the surface. 9a Given a non-multiple degenerate surface of the second order.

Hence it is not the case that both invariant points can be contained in both invariant planes – in contradiction to (B). IV). V) Let all points of ε1ε3 be invariant and hence both ε1 and ε3 are of class (V). Let A4 be the center for the elation in ε1. Then the center for ε3, say A2, cannot coincide with A4, because if this were the case, all planes through A4 would be invariant planes whereas only the planes of the pencil ε1ε3 contain all invariant points – in contradiction to (B). 5) must coincide with the pencil of ε1ε3 in order to comply with (B).

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