ABSTRACT
Chapter 5 contains the study of inner product and Hilbert spaces, and properties of operators between Hilbert spaces. The theory of Hilbert spaces provides essential tools in the theories of partial differential equations, quantum mechanics, Fourier analysis including applications to signal processing, and thermodynamics. First, many standard examples are given and the more familiar results are established, such as the Cauchy–Schwarz inequality and the parallelogram law and the P. Jordan–J. von Neumann theorem. Also the concepts of orthogonality and orthonormality are introduced, and the theorem of the elements of minimal norm, the theorem on the orthogonal decomposition, the Riesz representation theorem, Bessel’s inequality, Fourier coefficients, Parseval’s equality and the Gram–Schmidt orthogonalization process are proved. Additional topics are Hilbert adjoint and Hermitian, normal, positive and unitary operators. Finally, the chapter contains a detailed study of projectors and orthogonal projectors, and formulas and results on the norm of idempotent operators.
