Inner Product Spaces

Harkrishan Lal Vasudeva ()
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Harkrishan Lal Vasudeva: Indian Institute of Science Education and Research

Chapter Chapter 2 in Elements of Hilbert Spaces and Operator Theory, 2017, pp 21-151 from Springer

Abstract: Abstract This Chapter includes detailed study of inner product spaces and their completions. The space $$ L^2(X,\rm{\mathfrak{M}},\rm{\mu}) $$ , where $$ X,\rm{\mathfrak{M}},\rm{\mu} $$ denote respectively a nonempty set, a $$ \rm{\sigma} $$ -algebra of subsets of X and an extended nonnegative real-valued measure has been studied; so is the space $$ A(\rm{\Omega}) $$ of holomorphic functions on a bounded domain $$ \rm{\Omega} $$ in $$ \mathbb{C} $$ . These spaces are some of the important examples of Hilbert spaces. Included here are many applied topics such as Legendre, Hermite, Laguerre polynomials, Rademacher functions and Fourier series. Linear functional on Hilbert spaces and the related Riesz Representation Theorem have been described. Applications of Hilbert space theory to diverse branches of mathematics are included too.

Date: 2017
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DOI: 10.1007/978-981-10-3020-8_2

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