Hilbert Spaces

Abstract This chapter explores the basics of Hilbert spaces by using n-dimensional Euclidean space, he space of square-summable complex sequences, and the space of square-integrable, complex-valued Lebesgue-measurable function as examples. In addition, this chapter covers the Cauchy–Schwarz and triangle inequalities, orthonormal bases, and orthogonal projections. Since Banach spaces play a role in the subsequent chapters, this chapter also covers a few Banach-space basics.