For any Banach space $X$ one has a family of semi-norms: $$\{\|\cdot\|_{x^*}\mid x^*\in X^*\}\qquad \|x\|_{x^*}=|x^*(x)|$$ these semi-norms induce the weak topology on $X$. But they also induce a uniform structure.
Are Banach spaces with this weak uniform structure complete?
No. For an example, let $X=\ell^2$ and consider the sequence $x_n=(1,\ldots,1,0,0,\ldots)$ with $n$ entries equal to $1$. This is a weak Cauchy sequence which does not converge.
As far as I remember, only finite-dimensional Banach spaces can be weakly complete.