I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me deduce how these conclusions are drawn since I can't; I have a few areas of confusion, I hope that's ok. The content from the notes is italicized below.
With regards to topological vector spaces: 'Metrizable' means the topology is given by a metric. We seldom use this directly; instead, we use the fact that there is a decreasing base $(N_j)_{j=1}^{\infty}$ of closed, absolutely convex neighbourhoods of $0$. If $p_j$ is the gauge (Minkowski functional) of $N_j$ then the topology is determined by the increasing sequence $(p_j)_{j=1}^{\infty}$ of semi-norms. We could then take for example the metric $\rho(x,y) = \sum_{j=1}^{\infty} \frac{p_j(x-y)}{2^j(p_j(x-y)+1)}$.
Note that $x_n \to x$ as $n \to \infty $ iff $p_j(x_n-x) \to 0 $ as $n \to \infty $, for each $j \in \mathbb{N} $. Here are 2 easy examples: first, let $ \omega$ be the space of all complex sequences. Set $p_j(x) = j \sum_{i=1}^j |x_i| $; then the topology on $\omega$ is simply the product topology. Secondly, let $C(\mathbb{R}^d)$ be the space of all continuous functions on $\mathbb{R}^d$. Set $p_j(f) = j\, \sup_{|x| \leq j} |f(x)| $. The resulting topology is the topology of local uniform convergence.
So, my main confusion is how we conclude in the 2 examples that the topology is precisely the product/local uniform convergence topology? Are we defining seminorms $p_j$ (with the factor of $j$ at the start to ensure they're increasing), and then saying 'these must be the Minkowski functionals for some decreasing base of neighbourhoods of $0$', then using the fact that convergence under the suggested metric occurs only if convergence occurs for each $p_j$ to see what a limit is under in this topology to deduce what the whole topology is? So, say in the fist case we deduce that for convergence to $0$, every finite sum $ \sum_{i=1}^j |x^{(n)}_i| $ of terms of the sequence $(x^{(n)})_{n=1}^\infty$ tends to $0$, then perhaps there's some way I can't see that this type of limit behaviour implies the product topology.
For another thing, while it's obvious that given some set we may induce a Minkowski functional by its very definition, it's not obvious to me that conversely by defining a seminorm $p_j$, we induce some unique neighbourhood for which it is the Minkowski functional: this is a smaller issue however. Despite the fact these are meant to be 'easy' examples, I simply can't see why this collection of seminorms induces precisely the topology stated, in either case. Could anyone explain it to me? I would ask my lecturer but since I wasn't able to attend the course which is now over (it clashed with another) it may be difficult to clear up my confusion. Thank you in advance.