An Extension of Polyak’s Theorem in a Hilbert Space

Abstract

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅) _H . Let us consider three linear bounded operators,

$$A_{i}:H\rightarrow H,\quad\,i=1,2,3.$$

We define the functions

$$\begin{array}{rcl}\varphi_{i}(x)&=&(A_{i}x,x)_{H}+2(a_{i},x)_{H}+\alpha_{i},\quad\forall x\in H,\ i=1,2,\\[3pt]f_{i}(x)&=&(A_{i}x,x)_{H},\quad\forall x\in H,\ i=1,2,3,\end{array}$$

where a _i ∈H and α _i ∈ℝ. In this paper, we discuss the closure and the convexity of the sets Φ _H ⊂ℝ² and F _H ⊂ℝ³ defined by

$$\begin{array}{rcl}\Phi_{H}&=&\{(\varphi_{1}(x),\varphi_{2}(x))\mid x\in H\},\\[3pt]F_{H}&=&\{(f_{1}(x),f_{2}(x),f_{3}(x))\mid x\in H\}.\end{array}$$

Our work can be considered as an extension of Polyak’s results concerning the finite-dimensional case.