Yu. G. Reshetnyak

Stability Theorems in Geometry and Analysis - Mathematics and Its Applications Yu. G. Reshetnyak 1st Ed. Softcover of Orig. Ed. 1994 edition

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.