Suppose I want to construct an orthonormal basis for a vector space with "almost" equal magnitude components. I.e. for 2D I can do this exactly:
$\vec{v}_1 = \frac{1}{\sqrt{2}}(1, 1)^T $ and $\vec{v}_2 = \frac{1}{\sqrt{2}}(1, -1)^T $
2D is a special case though. In general I can't do this so all components are exactly equal. Is there a procedure I could follow to do as good as I can for arbitrary dimension? Can I bound the size of the maximum component of any vector in the basis to be less than or equal to $O(1/\sqrt{d})$ for a $d$ dimensional vector space over the reals?
Thanks!