note that it is possible to avoid the explicit construction of the lower triangular matrix in ( via the following modification of algorithm which only stores an extra vector at each recursive step of the levinson-durbin algorithm algorithm solving an xn bn with (nmemory cost inputfirst row [ > - of matrix an and right-hand side bn outputsolution xn - bn for do [xk xk- [yk yk- > (bk+ > )/ ay (ak+ > )/ [ xa ] [ ay yay ] return function space functional analysis much of the previous theory on euclidean vector spaces can be generalized to vector spaces of functions every element of (real-valuedfunction space is function from some set to rand elements can be added and scalar multiplied as if they were vectors in other wordsif and hthen bg for all ab on we can impose an inner product as mapping ** from to that satisfies ha gi ah gi bh gi fgi hgf ff ff if and only if (the zero functionnorm complete we focus on real-valued function spacesalthough the theory for complex-valued function spaces is similar (and sometimes easier)under suitable modifications ( fgi hgf isimilar to the linear algebra setting in section we say that two elements and in are orthogonal to each other with respect to this inner product if fgi given an inner productwe can measure distances between elements of the function space using the norm : ff for examplethe distance between two functions fm and fn is given by fm fn the space is said to be complete if every sequence of functions for which fm fn as mn (