Bounded linear operators in Banach and Hilbert spaces

Abstract

The initial and foundational category of spaces examined in functional analysis is that of complete normed spaces over real or complex numbers, known as Banach spaces. A notable example is a Hilbert space, in which the norm is derived from an inner product. These spaces are pivotal in diverse fields, such as the mathematical representation of quantum mechanics, machine learning, partial differential equations, and Fourier analysis. This dissertation extensively investigates continuous linear operators de- fined on Banach and Hilbert spaces. It delves into crucial theorems and properties of bounded linear operators, including the Hilbert-adjoint operators. Consequently, the Hilbert-adjoint operator is used to define three essential classes of bounded linear operators on a Hilbert space. The study concludes by examining the spectral properties of bounded self-adjoint linear operators. Key Words: Normed spaces, Banach spaces, Hilbert spaces, Bounded linear operators, Spectral theory.