We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0versus alternatives $H_{\epsilon ,\tau }:v\in V_{\epsilon }$ for the sets $V_{\epsilon }=V_{\epsilon }(\tau ,{\rho }_{\epsilon })\subset l_2$. The sets Vε are lq-ellipsoids of semi-axes ai = i-s R/ε with lp-ellipsoid of semi-axes bi = i-r pε/ε removed or similar Besov bodies Bq,t;s (R/ε) with Besov bodies Bp,h;r (pε/ε) removed. Here $\tau =(\kappa ,R)$ or $\tau =(\kappa ,h,t,R);\kappa =(p,q,r,s)$ are the parameters which define the sets Vε for given radii...