Hilbert spaces of vector-valued functions generated by quadratic forms

We study Hilbert spaces L 2 ( E , G ) , where E ⊂ R d is a measurable set, | E | > 0 and for almost every t ∈ E the matrix G ( t ) (see (3)) is a Hermitian positive-definite matrix. We find necessary and sufficient conditions for which the projection operators T k ( f ) ( ⋅ ) = f k ( ⋅ ) e k , 1 ≤ k ≤ n are bounded. The obtained results allow us to translate various questions in the spaces L 2 ( E , G ) to weighted norm inequalities with weights which are the diagonal elements of the matrix G ( t ) . In Section 3 we study the properties of the system { φ m ( t ) e j , 1 ≤ j ≤ n ; m ∈ N } in the space L 2 ( E , G ) , where Φ = { φ m } m = 1 ∞ is a complete orthonormal system defined on a measurable set E ⊂ R . We concentrate our study on two classical systems: the Haar and the trigonometric systems. Simultaneous approximations of n elements F 1 , … , F n of some Banach spaces X 1 , … , X n with respect to a system Ψ which is a basis in any of those spaces are studied.