The paper by Laurent and Massart addresses the problem of estimating the norm squared of a function \( s \) in a separable Hilbert space \( \mathbb{H} \) when observations are given by a Gaussian process \( Y(t) = \langle s, t \rangle + \sigma L(t) \), where \( L \) is a Gaussian isonormal process. The authors propose a model selection approach using penalized least squares to build estimators of \( \|s\|^2 \). They prove a nonasymptotic risk bound for the penalized estimator, showing that it is adaptive over various collections of sets for the parameter \( s \), including hyperrectangles, ellipsoids, \( l_p \)-bodies, and Besov bodies. The choice of the penalty function is crucial and depends on the complexity of the model collection. The paper also discusses the efficiency of the estimator when the noise level \( \sigma \) tends to zero and provides computational methods for the penalized estimator. The results are demonstrated through examples in the context of the Gaussian sequence model, where the authors show that their estimator is efficient and adaptive over a wide range of parameter sets.The paper by Laurent and Massart addresses the problem of estimating the norm squared of a function \( s \) in a separable Hilbert space \( \mathbb{H} \) when observations are given by a Gaussian process \( Y(t) = \langle s, t \rangle + \sigma L(t) \), where \( L \) is a Gaussian isonormal process. The authors propose a model selection approach using penalized least squares to build estimators of \( \|s\|^2 \). They prove a nonasymptotic risk bound for the penalized estimator, showing that it is adaptive over various collections of sets for the parameter \( s \), including hyperrectangles, ellipsoids, \( l_p \)-bodies, and Besov bodies. The choice of the penalty function is crucial and depends on the complexity of the model collection. The paper also discusses the efficiency of the estimator when the noise level \( \sigma \) tends to zero and provides computational methods for the penalized estimator. The results are demonstrated through examples in the context of the Gaussian sequence model, where the authors show that their estimator is efficient and adaptive over a wide range of parameter sets.