Limits on locally convex spaces

Question

A curve on a locally convex space is a function $\gamma : I \to F$ where $F$ is a locally convex space and $I \subseteq \mathbb{R}$ is an interval. The curve is differentiable if the following limit exists: $$ \gamma'(x) := \lim_{t \to 0}\frac{\gamma(x+t)-\gamma(x)}{t} $$ but what does this limit mean? I mean...elements $\gamma(x+t)$ and $\gamma(x)$ are in a lcs and this is not (necessarily) a normed space. I'm really stuck at this definition.

If $F$ is locally convex, then it is a topological vector space (with, say, a topology given by a family of seminorms). The notion of a limit is replaced by the following.

Definition: Let $f: I \subseteq \mathbb{R} \to F$. We write $\lim_{x \to a}f(x) = L$ if for every neighborhood $V$ of the origin there exists $\delta > 0$ such that $0 \lt |x-a| \lt \delta$ implies $f(x) - L \in V$.

Is this the right definition?

Answer 1

As explained by hardmath, the short answer is: yes this is the correct definition. It is just what one means by limit with values in a topological space which includes as a particular case LCTVS's. More useful in practice are the following equivalent conditions

1. For all continuous seminorm $||\cdot||$ on $F$, $\lim_{x\rightarrow a}||f(x)-L||=0$.
2. For all seminorm $||\cdot||$ in $\mathscr{A}$, $\lim_{x\rightarrow a}||f(x)-L||=0$.

Here $\mathscr{A}$ is your favorite set of seminorms on $F$ which define the given locally convex topology on $F$.