According to the postulates of quantum mechanics, the state of a physical system, at fixed time , is given by the wave function belonging to a separable complex Hilbert space. As such, the inner product of two wave functions and can be defined as the complex number (at time )The functions are here assumed to be elements of , the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of Lebesgue measure . This is necessary to obtain an inner product (that is, ) as opposed to a semi-inner product. The integral is taken to be the Lebesgue integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space. .
ACcording to the postUlaTEs of QUAntum mechanics, ThE State oF a phySicAl sysTeM, aT FiXed tIme , is given BY tHe waVe fUnctIoN BeLonging to a SepARAble cOmPLex HILbERt spAcE. As sUcH, The inner proDUcT of tWO wAvE funcTionS And caN BE definEd as The CoMplex nUmbER (at Time )The fUnctIonS aRe HerE asSumed tO BE elEmEnts of , the SpaCe oF SQuAre inTegrAble fuNCtiONs. ThE eLemenTs of this sPAce Are MOre preciseLy equivaLence cLAsseS oF SquArE INteGRaBle fUncTions, TwO funcTions decLAred EqUivaleNt If tHeY DIfFeR oN a Set oF leBesGue meASurE . this iS necESsary To obTaIn aN inner pRoduCT (that is, ) as opPoseD TO A seMi-iNner pROduCt. The inTegral Is taken tO Be tHe lebeSgue IntEgrAl. this is EsSEntIal fOr cOMPLeteness oF tHE space, thus yIELding a ComPLete iNner ProdUCt space = HIlbert spaCe. .