We consider the problem of estimating $\|s\|^2$ when $s$ belongs
to some separable Hilbert space and one observes the Gaussian process $Y(t) =
\langles, t\rangle + \sigmaL(t)$, for all $t \epsilon \mathbb{H}$,where $L$ is
some Gaussian isonormal process. This framework allows us in particular to
consider the classical “Gaussian sequence model” for which
$\mathbb{H} = l_2(\mathbb{N}*)$ and $L(t) =
\sum_{\lambda\geq1}t_{\lambda}\varepsilon_{\lambda}$, where
$(\varepsilon_{\lambda})_{\lambda\geq1}$ is a sequence of i.i.d. standard
normal variables. Our approach consists in considering some at most countable
families of finite-dimensional linear subspaces of $\mathbb{H}$ (the
models) and then using model selection via some conveniently penalized
least squares criterion to build new estimators of $\|s\|^2$. We prove a
general nonasymptotic risk bound which allows us to show that such
penalized estimators are adaptive on a variety of collections of sets for the
parameter $s$, depending on the family of models from which they are built.In
particular, in the context of the Gaussian sequence model, a convenient choice
of the family of models allows defining estimators which are adaptive over
collections of hyperrectangles, ellipsoids, $l_p$-bodies or Besov bodies.We
take special care to describe the conditions under which the penalized
estimator is efficient when the level of noise $\sigma$ tends to zero. Our
construction is an alternative to the one by Efroïmovich and Low for
hyperrectangles and provides new results otherwise.