This paper presents a non-asymptotic risk bound for adaptive estimation of the quadratic functional $\|s\|^2$ in a Gaussian process framework. The authors consider a separable Hilbert space $H$ and a Gaussian process $Y(t) = \langle s, t \rangle + \sigma L(t)$, where $L$ is a Gaussian isonormal process. They propose a model selection approach using penalized least squares to build adaptive estimators of $\|s\|^2$. The estimators are shown to be adaptive over various classes of parameter sets, including hyperrectangles, ellipsoids, $l_p$-bodies, and Besov bodies. The method is demonstrated in the context of the Gaussian sequence model, where the noise level $\sigma$ is set to $\sigma = n^{-1/2}$. The authors derive a general non-asymptotic risk bound that allows them to show the adaptivity of their estimators. They also discuss the efficiency of the penalized estimator as the noise level tends to zero. The results are compared to previous methods, showing that their approach provides improved performance in certain cases. The paper also addresses the adaptivity of the estimator over different classes of parameter sets, including $l_p$-bodies for $p \geq 2$, and provides uniform risk bounds for the penalized estimator over these sets. The results are illustrated with examples in the Gaussian sequence model, showing that the estimator is adaptive over various classes of parameter sets and achieves optimal rates of convergence in certain cases.This paper presents a non-asymptotic risk bound for adaptive estimation of the quadratic functional $\|s\|^2$ in a Gaussian process framework. The authors consider a separable Hilbert space $H$ and a Gaussian process $Y(t) = \langle s, t \rangle + \sigma L(t)$, where $L$ is a Gaussian isonormal process. They propose a model selection approach using penalized least squares to build adaptive estimators of $\|s\|^2$. The estimators are shown to be adaptive over various classes of parameter sets, including hyperrectangles, ellipsoids, $l_p$-bodies, and Besov bodies. The method is demonstrated in the context of the Gaussian sequence model, where the noise level $\sigma$ is set to $\sigma = n^{-1/2}$. The authors derive a general non-asymptotic risk bound that allows them to show the adaptivity of their estimators. They also discuss the efficiency of the penalized estimator as the noise level tends to zero. The results are compared to previous methods, showing that their approach provides improved performance in certain cases. The paper also addresses the adaptivity of the estimator over different classes of parameter sets, including $l_p$-bodies for $p \geq 2$, and provides uniform risk bounds for the penalized estimator over these sets. The results are illustrated with examples in the Gaussian sequence model, showing that the estimator is adaptive over various classes of parameter sets and achieves optimal rates of convergence in certain cases.