In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$.
In the vector space context, the term local base will always mean a local base at $0$. A local base of a topological vector space $X$ is thus a collection $\mathfrak{B} $ of neighborhoods of $0$ such that every neighborhood of $0$ contains a member of $\mathfrak{B} $. The open sets of $X$ are then precisely those that are unions of translates of members of $\mathfrak{B} $.
After that, we have definitions of few spaces, among them there are:
(a) $X$ is locally convex if there is a local base $\mathfrak{B} $ whose members are convex.
...
(e) $X$ is an $F$-space if its topology $\tau$ is induced by a complete invariant metric $d$. (Compare Section 1 .25.)
(f) $X$ is a Frechet space if $X$ is a locally convex $F$-space.
But the problem is, I don't really see the difference in spaces e) and f) presented above.
Let's say that we have topological space $(X,d)$, which topology $\tau$ is induced by a complete invariant metric $d$. Let's consider any ball with center in $0$ and radius $r$, which is
$$B(0,r) = \{ x \in X : d(0,x) < r \} .$$
Now, if we have that ball's are convex
$$ t B(0,r) + (1 - t)B(0,r) \subseteq B(0,r) \quad \quad \quad for \quad (0 < t < 1) $$
then because local base is collection of neighborhoods of $0$ and we can obtain any open set just by translations, then $(X,d)$ is a locally convex $F$-space, thus Frechet space. But, can these ball be not convex? I thought they should, or at least I didn't recall to met such examples yet, but then there would be no difference between these spaces (if I am correct).
Summing it up, can someone help me with this matter and tell something more about balls that are not convex? Of course if this is what's the problem here - if I made some error and it's not about balls being convex or not, and there are more differences, then I would be glad for any explanation and advice.
A standard example of an F-space which is not locally convex is the $L^p$ space with $0 < p < 1$. Let $(X,\mu)$ be your favorite measure space and let $L^p(\mu)$ be the space of all (equivalence classes of) measurable functions with $\int |f|^p\,d\mu\ < \infty$. Equip it with the metric $d(f,g) = \int |f-g|^p\,d\mu$. (Note we do not have a $1/p$ power on the outside, so this is not a norm as it fails to be homogeneous; but if we did include the $1/p$ the triangle inequality would fail.) Unless $(X,\mu)$ is something silly like a finite set, the resulting topological vector space is not locally convex.
You can find some details in these lecture notes by Keith Conrad, especially example 2.19. The only part he doesn't discuss is the completeness, but this goes in the same way as the completeness of $L^p$ for $p \ge 1$.