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The following theorem is the analogue in our language of the previous fact about inner product spaces.
\label{thm:ort}
Fix \(\Gamma=(\Gamma_{\mathfrak{R}}\), \(\Gamma_{\mathfrak{A}})\) a pair of growth spaces of rank 2 and \(\mathcal{E}\) an error space of rank 2. Assume there is \(\zeta: {\mathbb{N}}^2 \rightarrow {\mathbb{R}}^{>0}\) s.t. \(\zeta \in \mathcal{E}\) and \({\lfloor \log \max(-\log \zeta, 2) \rfloor} \in \Gamma_{\mathfrak{A}}\)1. Consider \((\mu,f)\) a distributional estimation problem and \(P\) an \(\mathcal{E}(\Gamma)\)-optimal predictor for \((\mu,f)\). Then, \(P\) is also an \(\mathcal{E}^{\frac{1}{2}\sharp}(\Gamma)\)-optimal predictor for \((\mu,f)\).
If \({\lfloor \log(k+2) \rfloor}, {\lfloor \log(j+2) \rfloor} \in \Gamma_{\mathfrak{A}}\) (equivalently \(\Gamma_{\text{log}}^2 \subseteq \Gamma_{\mathfrak{A}}\)) then this condition holds for any \(\mathcal{E}\) since we can take \(\zeta = 2^{-h}\) for \(h\) polynomial.↩