I'm reading Dirac's "Principles of QM" and I reached the point (page 39), when he introduces the concept of expressing the scalar product of two kets using the complete set of eigenkets of an observable $\xi$. So, as I understood, he take into exam the simpler scenario where a ket can be expressed only using a complete and continuos set of eigenkets of $\xi$, excluding the hypothesis of some discrete eigenkets outside this interval. Therefore he writes two arbitrary kets as:
$$ | X \rangle = \int |\xi'x \rangle d\xi'$$ $$ | Y \rangle = \int |\xi''y \rangle d\xi''$$
He then says that it is possible to calculate the scalar product $\langle X | Y \rangle$ by taking the conjugate imaginary (his words) of the first:
$$ \langle X | Y \rangle = \iint \langle \xi'x | \xi''y \rangle d\xi' d\xi'' $$
He says that taking only $\int \langle \xi'x | \xi''y \rangle d\xi'' $, the result vanishes for every eigenket except for the one associated with $\xi' = \xi ''$ for the orthogonality theorem. He concludes that for this reason also $\langle X | Y \rangle$ vanishes since the above conclusion holds for every $\xi'$. And then it comes the part I was struggling about: Dirac says that normally $\langle X | Y \rangle$ doesn't vanish, "so in general $\langle \xi'x | \xi''y \rangle$ must be infinitely great in such a way as to make $\int \langle \xi'x | \xi''y \rangle d\xi' d\xi''$ non-vanishing and finite" (page 39 in my edition). I really couldn't understand this part: I might think that this way to make this integral finite would be the Dirac's delta. But I don't understand how making $\langle \xi'x | \xi''y \rangle$ infinitely great could help fixing the integral...
The same situation occurs after, when Dirac's discuss the case of $|X \rangle$ and $| Y \rangle$ are equal: he, in fact, says that in general the result is $\langle \xi'x | \xi'y \rangle$