Chapter 1: Functional Analysis for Imaging

Learning Objectives

• Define Banach and Hilbert spaces and verify completeness for the $L^p$ family
• State and prove the Riesz representation and orthogonal projection theorems
• Characterize bounded, compact, and adjoint operators and compute their norms
• Decompose a compact operator via its singular system and state the Picard condition
• Work with distributions, Sobolev spaces, and Green's functions in the PDE sense
• Formulate an imaging problem as a compact forward operator between Hilbert spaces