Bounded Linear Operators in Hilbert Spaces
Edoardo Provenzi · 2021
This chapter begins by introducing formal definitions for continuous and bounded operators. The continuity of a linear operator is equivalent to sequential continuity. The chapter focuses on bounded linear operators on Hilbert spaces, it is important to show at least one example of a non-bounded linear operator. The Riesz representation theorem owes its name to the fact that it allows all continuous linear functions on a Hilbert space to be represented via inner product. The Riesz representation theorem is one of the most important results of functional analysis. The Lax-Milgram theorem is widely used in solving partial differential equations expressed in variational form. The chapter examines the concept of orthogonal projection in a Hilbert space. It aims to determine the properties of isometric and unitary operators in a Hilbert space of infinite dimension, and provides an algebraic and geometric characterization of these operators.