# Angles of a triangle, in Hilbert spaces and in metric spaces

Solid angles of a tetrahedron

Does the statement

"the side opposite the largest angle has the largest length (and similarly for smallest angle)"

hold in a) Hilbert spaces? b) metric spaces? (where angle is defined by the construction in: A new notion of angle between three points in a metric space by Andrea Mondino http://arxiv.org/abs/1302.0520), c) any other spaces where angles and lengths can be defined? d) in rational trigonometry the statement would be about spreads and quadrances in any field with any quadratic form, but how would you order the spreads and quadrances of a triangle in an arbitrary field?

In b) we have metric but no algebra, and in d) we have algebra but no metric.

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In a Hilbert space, each triangle lies in a 2-dimensional subspace where lengths and angles are exactly as in Euclidean $\mathbb R^2$, so Euclidean facts about triangles remain true for triangles in Hilbert space. –  Andreas Blass Mar 19 '13 at 12:53