converges to some hthat isk fn as sequence that satisfies ( is called cauchy sequence complete inner product space is called hilbert space the most fundamental hilbert space of functions is the space an in-depth introduction to requires some measure theory [ for our purposesit suffices to assume that rd and that on measure is defined which assigns to each suitable set positive number ( ( its volumein many cases of interest is of the form (aw(xdx ( cauchy sequence hilbert space measure where is positive function on which is called the density of with respect to the lebesgue measure (the natural volume measure on rd we write (dxw(xdx to indicate that has density another important case is where (aw( )( density azd where is again called the density of ubut now with respect to the counting measure on zd (which counts the points of zd integrals with respect to measures in ( and ( can now be defined as (xu(dxf (xw(xdxand (xu(dxx (xw( ) respectively we assume for simplicity that has the form ( for measures of the form ( (so-called discrete measures)replace integrals by sums in what follows definition space let be subset of rd with measure (dxw(xdx the hilbert space (xuis the linear space of functions from to that satisfy ( ) (xdx ( and with inner product fgi if (xg(xw(xdx ( let be hilbert space set of functions fi iis called an orthonormal system not all sets have measure suitable sets are borel setswhich can be thought of as countable unions of rectangles orthonormal system