A real inner product space (X, δ) is a real vector space X together with a mapping δ: X × X → ℝ satisfying
  1. (i)


  2. (ii)


  3. (iii)


  4. (iv)

    δ(x,x)>0 for x≠0

for all x, y, zX and λ ∈ ℝ. Concerning the notation δ: X × X → ℝ and others we shall use later on, see the section Notation and symbols of this book. Instead of δ (x, y) we will write xy or, occasionally, x · y. The laws above are then the following: xy=yx,(x+y)z=xz+yz,(⋋x)⋅y=⋋⋅(xy) for all x, y, zX, λ ∈ ℝ, and x2:= x · x > 0 for all xX{0}. Instead of (X, δ) we mostly will speak of X, hence tacitly assuming that X is equipped with a fixed inner product, i.e. with a fixed δ: X × X → ℝ satisfying rules (i), (ii), (iii), (iv).


Distance Function Rational Number Product Space Real Vector Space Hyperbolic Geometry 
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© Birkhäuser Verlag AG 2007

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