Abstract
Real and complex inner product spaces are defined and several examples are studied. Elementary properties of inner products, such as the Cauchy–Schwarz–Bunyakovsky Theorem and Minkowski’s inequality are proven. The Lagrange identity relating inner and cross products in three-dimensional real vector spaces is proven. Normed spaces are defined and various examples of norms are considered, including spectral norms and various matrix norms. The Hahn–Banach Theorem, Gershgorin’s Theorem, and the Diagonal Dominance Theorem are proven. Matrix exponentials are studied.
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- 1.
This theorem was proven by the French mathematicians L. Lévy and J. Desplanques at the end of the nineteenth century. It was independently rediscovered by several other algebraists, including Hadamard, Minkowski, and Nekrasov.
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© 2012 Springer Science+Business Media B.V.
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Golan, J.S. (2012). Inner Product Spaces. In: The Linear Algebra a Beginning Graduate Student Ought to Know. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2636-9_15
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DOI: https://doi.org/10.1007/978-94-007-2636-9_15
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2635-2
Online ISBN: 978-94-007-2636-9
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