ABSTRACT
This chapter introduces an important class of spaces namely, the Hilbert spaces, which are a special class of metric spaces. It describes the Cauchy-Schwartz inequality and orthogonal complement in Hilbert spaces. The chapter studies separable Hilbert spaces and the related orthonormal bases. It discusses Bessel inequality showing that every linearly ordered subset has an upper bound, so that by Zorn’s lemma the collection of all orthonormal sets in Hilbert space has a maximal element, that is, an orthonormal set not properly contained in any other orthonormal set. The chapter also describes the Gram-Schmidt orthonormalization and proves that a Hilbert space is separable if and only if it has a countable orthonormal basis. It presents the statement and proof of two well-known results, namely, the Stampacchia and Lax-Milgram theorems.
