An Extension of Polyak’s Theorem in a Hilbert Space

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅)H. Let us consider three linear bounded operators, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{i}:H\rightarrow H,\quad\,i=1,2,3.$$\end{document} We define the functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rcl}\varphi_{i}(x)&=&(A_{i}x,x)_{H}+2(a_{i},x)_{H}+\alpha_{i},\quad\forall x\in H,\ i=1,2,\\[3pt]f_{i}(x)&=&(A_{i}x,x)_{H},\quad\forall x\in H,\ i=1,2,3,\end{array}$$\end{document} where ai∈H and αi∈ℝ. In this paper, we discuss the closure and the convexity of the sets ΦH⊂ℝ2 and FH⊂ℝ3 defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{rcl}\Phi_{H}&=&\{(\varphi_{1}(x),\varphi_{2}(x))\mid x\in H\},\\[3pt]F_{H}&=&\{(f_{1}(x),f_{2}(x),f_{3}(x))\mid x\in H\}.\end{array}$$\end{document} Our work can be considered as an extension of Polyak’s results concerning the finite-dimensional case.