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