[about this theorem:]
Any isometry of R^n that fixes a k-dimensional subspace is the product of at most n-k reflections.
[Dan's proof:]
The proposition is equivalent to saying that any isometry of R^p with no nontrivial fixed subspace is the product of at most p reflections.
Any isometry of R^n preserves mutually orthogonal subspaces: of dim = 2, and another one of dim = 1 (if n is odd). (This follows readily from complexification.)
The cases of 1 dimension, and one reflection in 2D, are trivial.
And a rotation of R^2 of angle theta is the product of a reflection about the x-axis, and one about the line at angle theta/2. QED.
Looks OK to me, but Fred's question wasn't "can someone give me a proof of this?" but "can someone explain the proof of this found in Conway&Smith's book". :-) -- g