by the recurrence ( )gn+ ( ( )gn (xngn- ( ) with ( and ( xfor the hermite polynomials are obtained when using instead the density of the standard normal distributionw(xe- / px these polynomials satisfy the recursion gn+ (xxgn (xdgn (xdx hermite polynomials with ( note that the hermite polynomials as defined above have not been normalized to have norm to normalizeuse the fact that kgn nwe conclude with number of key results in functional analysis the first one is the celebrated cauchy-schwarz inequality cauchyschwarz theorem cauchy-schwarz let be hilbert space for every fg it holds that | fgi kgk proofthe inequality is trivially true for (zero functionfor we can write ag hwhere and fgi/kgk consequentlyk | | kgk khk | | kgk the result follows after rearranging this last inequality let and be two linear vector spaces (for examplehilbert spaceson which norms kv and kw are defined suppose is mapping from to when vsuch mapping is often called an operatorwhen it is called functional mapping is said to be linear if ( bgaaf ba(gin this case we write instead of af if there exists such that ka kw kv ( then is said to be bounded mapping the smallest for which ( holds is called the norm of adenoted by kak (not necessarily linearmapping is said to be continuous at if for any sequence converging to the sequence af )af )converges to af that isif vk gkv kaf ( )kw operator functional linear mapping ( if the above property holds for every vthen the mapping itself is called continuous theorem continuity and boundedness for linear mappings for linear mappingcontinuity and boundedness are equivalent prooflet be linear and bounded we may assume that is non-zero (otherwise the statement holds trivially)and that therefore kak taking /kak in ( now ensures that ka agkw kak gkv kak this shows that is continuous bounded mapping norm continuous mapping