Abstract
The previous chapter introduced the concept of the norm of a vector as a generalization of the idea of the length of a vector. However, the length of a vector in ℝ2 or ℝ3 is not the only geometric concept which can be expressed algebraically. If x = (x 1, x 2, x 3) and y = (y 1, y 2, y 3) are vectors in ℝ3 then the angle, θ, between them can be obtained using the scalar product (x, y) = x 1 y 1 + x 2 y 2 + x 3 y 3 = ∥x∥ ∥y∥ cos θ, where \( \left\| x \right\| = \sqrt {x_1^2 + x_2^2 + x_3^2} = \sqrt {\left( {x,x} \right)} \) and \(\left\| y \right\| = \sqrt {\left( {y,y} \right)} \) are the lengths of x and y respectively. The scalar product is such a useful concept that we would like to extend it to other spaces. To do this we look for a set of axioms which are satisfied by the scalar product in ℝ3 and which can be used as the basis of a definition in a more general context. It will be seen that it is necessary to distinguish between real and complex spaces.
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© 2000 Springer-Verlag London
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Rynne, B.P., Youngson, M.A. (2000). Inner Product Spaces, Hilbert Spaces. In: Linear Functional Analysis. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-3655-2_3
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DOI: https://doi.org/10.1007/978-1-4471-3655-2_3
Publisher Name: Springer, London
Print ISBN: 978-1-85233-257-0
Online ISBN: 978-1-4471-3655-2
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