converselysuppose is continuous in particularit is continuous at (the zeroelement of vthustake and let and be as in ( for any let /( kgkv as khkv / dit follows from ( that kahkw kagkw kgkv rearranging the last inequality gives kagkw /dkgkv showing that is bounded theorem riesz representation theorem any bounded linear functional ph on hilbert space can be represented as ph(hhhgifor some (depending on phprooflet be the projection of onto the nullspace of phthat isn { ph( if ph is not the -functionalthen there exists with ph( let pg then and ph( ph( take /ph( for any hf : ph( ) lies in as it holds that fg which is equivalent to hhg ph(hkg by defining /kg we have found our representation fourier transforms we will now briefly introduce the fourier transform before doing sowe will extend the concept of space of real-valued functions as follows definition space let be subset of rd with measure (dxw(xdx and [ then (xuis the linear space of functions from to that satisfy ( ) (xdx ( when (xuis in fact hilbert space equipped with inner product fgi (xg(xw(xdx ( we are now in position to define the fourier transform (with respect to the lebesgue measurenote that in the following definitions and we have chosen particular convention equivalent (but not identicaldefinitions exist that include scaling constants ( ) or ( )- and where - pt is replaced with pttor -