A note on quantitative stability in Hilbert spaces

Abstract

We study stability theory in Hilbert spaces quantitatively. We prove that the inner product on the unit ball is $(k,\epsilon)$-stable for all $k\ge \exp(\pi/\epsilon)$, and it is not $(k,\epsilon)$-stable for $k\le \exp(\log 2/\epsilon)$, showing that the growth is necessarily exponential in $1/\epsilon$. We then analyze how stability scales under nonlinear connectives applied to the inner product. In particular, for power-type predicates $f(x,y)=\langle x,y\rangle_+^\beta$ with $\beta<1$ we obtain upper and lower bounds of the form $\exp(C\epsilon^{-1/\beta})$, and for $\beta>1$ and integer powers $\langle x,y\rangle^d$ we retain the bilinear scale $\exp(C/\epsilon)$.