Bounded Linear Maps

In this chapter, the exploration of advanced linear algebra and functional analysis concepts unfolds, beginning with the notion of bounded linear maps, which elegantly combine linearity and boundedness, crucial in various mathematical applications. The concept of the adjoint operator is introduced, enabling the study of self-adjoint, normal, and unitary operators, each possessing distinct properties and widespread utility. Singular value decomposition (SVD) emerges as a powerful factorization method, revolutionizing linear equation solving. When standard matrix inverses do not exist, generalized inverses, such as the Moore–Penrose inverse, provide a flexible structure for solving systems of linear equations, enabling least square solutions to otherwise ill-posed problems in a number of mathematical and practical situations.