 However, it exists as a random generalized function, since for any formula_15 we have

hence

is a well-defined finite random number.Although the above argument shows that formula_27 does not exist as a random element of formula_11, it still could be that it is a random function on formula_13 in some larger function space.In fact, in dimension formula_30, an orthonormal basis of formula_31 is given by

and then formula_35 is easily seen to be a one-dimensional Brownian motion (or Brownian bridge, if the boundary values for formula_36 are set up that way).