#equation1816#
it must be that, for some u :
#equation1819#
Elements of equation (#tex2html_wrap_inline3053#) are of the linear space
(over #tex2html_wrap_inline3055#), consisting of all functions defined on #tex2html_wrap_inline3057# :
#equation1822#
Otherwise, they are linearly dependent and one of them can be formed
as a linear combination of the others.
#tex2html_wrap_inline3059#
In addition, #tex2html_wrap_inline3061# satisfies the <#629#> Weak Independence Property<#629#>
(WIP), if the above IP is only required to hold for all pairs with
#tex2html_wrap_inline3063#.
There are many nonlinear maps that do not satisfy IP. An example of
functions that belongs to this category are <#630#> polynomials<#630#>, <#631#>
periodic functions<#631#>, and the <#632#> exponential function<#632#>
[#4##1#].