Abstract
Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (., .)(H). Let us consider three linear bounded operators,
A(i) : H -> H, i = 1, 2, 3.
We define the functions
phi(i)(x) = (A(i)x, x)(H) + 2(a(i), x)(H) + alpha(i), for all x is an element of H, i = 1, 2, f(i)(x) = (A(i)x, x)(H), for all x is an element of H, i = 1, 2, 3,
where a(i) is an element of H and alpha(i) is an element of R. In this paper, we discuss the closure and the convexity of the sets Phi(H) subset of R(2) and F(H) subset of R(3) defined by
Phi(H) = {(phi(1)(x), phi(2)(x)) vertical bar x is an element of H}, F(H) = {(f(1)(x), f(2)(x), f(3)(x)) vertical bar x is an element of H}.
Our work can be considered as an extension of Polyak's results concerning the finite-dimensional case.