1. Preliminaries, Notation, and Terminology n n 1.1. Sets and
functions in lR. * Throughout the book, lR. stands for the
n-dimensional arithmetic space of points x = (X},X2,'" ,xn)j Ixl is
the length of n n a vector x E lR. and (x, y) is the scalar product
of vectors x and y in lR. , i.e., for x = (Xl, X2, *.* , xn) and y
= (y}, Y2,**., Yn), Ixl = Jx~ + x~ + ...+ x~, (x, y) = XIYl + X2Y2
+ ...+ XnYn. n Given arbitrary points a and b in lR. , we denote by
[a, b] the segment that joins n them, i.e. the collection of points
x E lR. of the form x = >.a + I'b, where>. + I' = 1 and >.
~ 0, I' ~ O. n We denote by ei, i = 1,2, ...,n, the vector in lR.
whose ith coordinate is equal to 1 and the others vanish. The
vectors el, e2, ...,en form a basis for the space n lR. , which is
called canonical. If P( x) is some proposition in a variable x and
A is a set, then {x E A I P(x)} denotes the collection of all the
elements of A for which the proposition P( x) is true.
General
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