chapter  4
27 Pages

Referential Duality and Representational Duality in the Scaling of Multidimensional and Infinite-Dimensional Stimulus Space: Jun Zhang

L1 Cr ~ x') == II J; - x'II . Such a Inetric is continuous with respect to (x, x') and it satisfies the axiOlns of (i) non-negativity: L1(x~ x') 2:: 0, with 0 attained if and only if x == x'; (ii) symmetry: L1(x, ~r/) == L1(x' , x); and (iii) triangle inequality: L1(x, x') + L1(x' , x") 2:: L1(x, x"), for any triplet J;, tr' , x". The norm II . II may also be used to define an inner product (., .):

The inner product operation on the vector space allows one to define the angle between two vectors (and hence orthogonality)~ which allows one to project one vector onto another (and hence onto a subspace).