The author in the book I am reading defines a locally convex space as follows.
Definition: A topological vector space $(X,\tau)$ is called a locally convex space if there is an index set $A$ and a family $\{p_\alpha : X \rightarrow [0,\infty), \alpha \in A \}$ of continuous seminorms, such that
(a) $\tau$ is the initial topology with respect to the canonical projections $\{ q_\alpha : X \rightarrow X/\ker(p_\alpha)\}_{\alpha\in A} $ onto the normed spaces $X/\ker(p_\alpha)$.
(b) If $\forall x\in X\ \forall \alpha\in A: p_\alpha(x)=0 $, then $x=0$. (This implies, that $\tau$ has the Hausdorff property.)
Then, the topology $\tau$ is called 'generated by the family $\{p_\alpha\}_{\alpha\in A}$' and the family of seminorms is called 'generating family'.
Later on, he defines a separating family of seminorms as follows.
Definition: A family of seminorms $\{p_\alpha\}_{\alpha\in A}$ on a vector space $X$ is said to be separating, if $\forall x\in X\ \exists \alpha\in A: p_{\alpha}(x)\ne 0$.
My confusion starts with the following Proposition.
Proposition: Let $X$ be a vector space and $\{p_\alpha\}_{\alpha\in A}$ a separating family of seminorms on $X$. Then a Hausdorff vector topology is generated by the subbase $$B_\epsilon^\alpha(x)=\{y\in X: p_\alpha(y−x)<\epsilon\},\ \alpha\in A,\ \epsilon>0,\ x \in X.$$ Thus, $(E,\{p_\alpha\}_{\alpha\in A})$ is a locally convex space and the topology contains a $0$-neighbourhood basis of convex sets. Finally, each $p_\alpha$ is continuous with respect to the locally convex topology.
Now, what confuses me the most is, that the author talks about a subbase. I think this is really a base for the initial topology of $\{ q_\alpha : X \rightarrow X/\ker(p_\alpha)\}_{\alpha\in A} $.
Proof: By using the fact, that $q_\alpha$ is surjective, a norm on $X/\ker(p_\alpha)$ can be defined through
$$\lVert q_\alpha(x)\rVert =\inf_{z\in\ker(p_\alpha)}p_\alpha (x+z).$$
Claim: $p_\alpha=\lVert q_\alpha \rVert$
Proof of Claim: Suppose $\inf_{z\in\ker(p_\alpha)}p_\alpha (x+z) < p_\alpha (x)$. Then there is a $z\in \ker(p_\alpha)$, such that $p_\alpha(x+z)<p_\alpha(x)$. Using the reversed triangle inequality, this yields $$0<p_\alpha(x)-p_\alpha(x+z)=|p_\alpha(x)-p_\alpha(x+z)|\le |p_\alpha(x-x-z)|=p_\alpha(z)=0,$$ which is contradictory. So it must hold $p_\alpha(x)\le \inf_{z\in\ker(p_\alpha)}p_\alpha (x+z)$. Since $0\in\ker(p_\alpha)$, it holds $\inf_{z\in\ker(p_\alpha)}p_\alpha (x+z)\le p_\alpha (x+0)=p_\alpha(x)$. In total, it holds $p_\alpha(x)=\lVert q_\alpha(x) \rVert$.
With this, one can see that the subbase is really a base: It is known, that the $\epsilon$-balls are a base for the topology on $[0,\infty)$ and that the preimage of a base is a base for the initial topology. That the subbase is in fact a base, should then follow from
$$B_\epsilon^\alpha(x)=\{y\in X: p_\alpha(y−x)<\epsilon\}=\{y\in X: \lVert q_\alpha(y−x)\rVert <\epsilon\}=q_\alpha^{-1}(B_\epsilon(q_\alpha(x))).$$
My question: I am just trying to make sense of this. In the context of locally convex spaces: Are there situations in which one is dealing with a subbase and situations in which one is dealing with a base? Is the Definition above slightly different from the usual definition of locally convex spaces? I do not come through...
Later on the author even states, that this subbase is a base, when it holds following property: For each two seminorms $p_\alpha$ and $p_\beta$, there is a seminorm $p_\gamma$ and $C > 0$, such that $\max\{p_\alpha(x),p_\beta(x)\}\le C p_\gamma(x)$ for all $x\in X$.
Any hint or help is appreciated! Thank you in advance!
