Let $X$ be a uniform space and $F(X)$ the vector space of all uniformly continuous real-valued functions over $X$. It is possible to express every bounded uniform semimetric $d$ on $X$ as $d(x,y) = d_B(x,y) := \sup_{f\in B} |f(x)-f(y)|$ with a suitable pointwise bounded and uniformly equicontinuous $B\subseteq F(X)$, choose e.g. $B = \{\,f_{\hat{x}}:X\rightarrow\mathbb{R}\,\,|\,\,\hat{x}\in X\,\}$ with $f_{\hat{x}}(x) := d(\hat{x},x)$.

In general, there should be more than only one uniform structure on $X$ that give rise to the same $F(X)$. The weakest one is clearly the weak uniform structure induced by $F(X)$ on $X$, i.e. the uniform structure given by the uniform semimetrics $d_{\{f\}}$ for all $f\in F(X)$. Is there also a strongest one (analogous to the Mackey-topology in the theory of locally convex spaces)?

Moreover, I would conjecture that (unlike for locally convex spaces) $X$ is complete if and only if it is complete under the weak uniform structure, because (unlike for locally convex spaces) the induced topologies are the same, namely the weak topology induced on $X$ by $F(X)$. Is this true?

I would also be very happy if someone could point out a good source for learning about uniform spaces that goes beyond a mere appendix to the theory of topological spaces.