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.
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Vasudeva, H.L. (2017). Inner Product Spaces. In: Elements of Hilbert Spaces and Operator Theory. Springer, Singapore. https://doi.org/10.1007/978-981-10-3020-8_2
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DOI: https://doi.org/10.1007/978-981-10-3020-8_2
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Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-3019-2
Online ISBN: 978-981-10-3020-8
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