I'm currently studying nuclear spaces and the nuclear spectral theorem with the books of Gelfand, Vilekin and Schilow, "Generalized functions", especially the second and fourth part. Currently, I'm trying to understand the proof, that every nuclear (or, more generally, every complete locally convex space with the Bolzano-Weierstraß-property, i.e. every bounded subset is precompact) is separable (to find in "Generalized Functions II", Chapter 1 §6.5).
The starting point is the statement $\Phi=\bigcap_{n\in\mathbb{N}}\Phi_n$ for a nuclear space $\Phi$ where each $\Phi_n$ is a Banach Space and $\|\phi\|_i\leq\|\phi\|_j$ for $i\leq j$ and therefore $\Phi_1\supset\Phi_2 \supset \dots$. This is all quite clear (note that the definitions may differ from other sources, e.g. the LCTVS/Nuclear Spaces carry a family of norms here, not just seminorms).
If every $\Phi_n$ is separable, so is $\Phi$, thats not too hard. So, if $\Phi$ wouldnt be separable, at least one $\Phi_n$ is not separable, wlog let this be $\Phi_1$. Gelfand&Schilow now state, that there exists a subset $Z_1\subset\Phi_1$ with $Z_1$ being uncountable, bounded in $\Phi_1$ and $\|\phi-\psi\|_1\geq \epsilon\;\forall\psi\neq\phi\in Z_1$. The book says this is just a consequence of the axiom of choice, but I am not able to prove that this statement is true. I tried to prove this by contradiction to the statement $\Phi_1$ is not separable by negating the properties stated, without success so far. Am I overlooking something very obvious?
The proof continues by inductively constructing $Z_n\subset\Phi_n$ with the same properties for every $n$ by just saying that while going from $n$ to $n+1$, we can just intersect $Z_n$ with some big ball in $\Phi_{n+1}$ and remain uncountable. I wasnt able to prove this statement either, I think it has to do with the unit balls being absorbing in $\Phi$. Any idea here?
The inductive limit of the $Z_n$ is then bounded in $\Phi$, but not precompact, which is a contradiction. But, as described above, some of the major steps remain unclear to me, I would be very thankful if anyone here could help me out.
Thanks in advance!