The book I'm reading on quantum mechanics states:
The eigenfunctions of an hermitian operator satisfy the completeness relation:
\begin{equation} \sum_n \psi^*_n(x')\psi_n(x) = \delta(x-x')\end{equation}
How is this definition of completeness related to the one defining a Banach space?
\begin{equation} \lim _{n\to \infty }\left\|x_{n}-x\right\|_{X}=0 \end{equation}
Or the relation it gives is not related to Banach spaces?
It is not the same type of completeness. Rather it refers to the "completeness" of an orthonormal basis $\psi_n$, in the sense of being a "total" set (Kreyszig terminology) or a "complete" set.
The Dirac "function" is not a classical function at all, but a generalized function, and is to be interpreted in the sense of $\int\delta(x-x')\phi(x')dx'=\phi(x)$. That is, in rigorous mathematics, the stated equation becomes $$\int\sum_n\psi^*_n(x')\psi_n(x)\phi(x')dx'=\phi(x)$$ valid for all $\phi\in L^2(\mathbb{R})$. This is equivalent to saying that in the Hilbert space $L^2(\mathbb{R})$, $$\sum_n\langle\psi_n,\phi\rangle\psi_n=\phi$$ which, in turn, is equivalent to the completeness of $\psi_n$.