quantum mechanics - Can eigenstates of a Hilbert space be thought of as delta functions?

Say we have an observable that describes a Hilbert space and that observable acts on state kets. Lets take the position observable for example. Then $\langle y|x\rangle = \delta(y - x)$. But can the eigenstates of the position observable be individually thought of as delta functions? $$ A |x\rangle = x'|x\rangle $$

Is this $|x\rangle$ then individiually a delta function picking $x'$ out of $A$? Wouldn't this also imply that we have an infinite number of delta function eigenstates in the observable space?