As it can be easily found in literature, indefinite integrals are treated as inner product of infinite dimensional vectors. But what about definite integrals? When we deal with definite integrals, I think, we would have been previously defined the associated vectors such that the integration limits to be applied on them. However, the vectors x in such infinite dimensional spaces are always shown by summation of xkak for k=1 to infinite. In better word, when we assume a given definite integral (with limits a, b) as an inner product of infinite dimensional vectors, how can we define the associated vectors? I will appreciate if anyone would help me.