Hilbert spaces of vector-valued functions generated by quadratic forms

We study Hilbert spaces L2(E,G)L2(E,G), where E⊂RdE⊂Rd is a measurable set, |E|>0|E|>0 and for almost every t∈Et∈E the matrix G(t)G(t) (see (3)) is a Hermitian positive-definite matrix. We find necessary and sufficient conditions for which the projection operators Tk(f)(⋅)=fk(⋅)ek,1≤k≤n are bounded. The obtained results allow us to translate various questions in the spaces L2(E,G)L2(E,G) to weighted norm inequalities with weights which are the diagonal elements of the matrix G(t)G(t). In Section 3 we study the properties of the system {φm(t)ej,1≤j≤n;m∈N} in the space L2(E,G)L2(E,G), where Φ={φm}m=1∞ is a complete orthonormal system defined on a measurable set E⊂RE⊂R. We concentrate our study on two classical systems: the Haar and the trigonometric systems. Simultaneous approximations of nn elements F1,…,FnF1,…,Fn of some Banach spaces X1,…,XnX1,…,Xn with respect to a system ΨΨ which is a basis in any of those spaces are studied.