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Inner Product Spaces, Hilbert Spaces

  • Bryan Patrick Rynne
  • Martin Alexander Youngson
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

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.

Keywords

Hilbert Space Orthonormal Basis Linear Subspace Product Space Orthogonal Complement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2000

Authors and Affiliations

  • Bryan Patrick Rynne
    • 1
  • Martin Alexander Youngson
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityRiccarton, EdinburghUK

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