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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?

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  • $\begingroup$ Here you say $F$ is a locally convex space, meaning that a topology is given based on neighborhoods of the origin which are absolutely convex absorbent sets (the notion can also be defined in terms of seminorms). So the $\epsilon-\delta$ definition of limit is replaced by one in which (instead of small $\epsilon \gt 0$) a small neighborhood of the origin is prescribed. $\endgroup$
    – hardmath
    Apr 7 '20 at 14:58
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    $\begingroup$ I think I got it. I'll edit my post with what I understood from your comment. $\endgroup$
    – MathMath
    Apr 7 '20 at 15:04
  • $\begingroup$ I did it. Is this what you meant? $\endgroup$
    – MathMath
    Apr 7 '20 at 15:14
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    $\begingroup$ Right. In the case of differentiability you apply that general notion of limit to get the meaning of the limit you give at the outset of your Question. $\endgroup$
    – hardmath
    Apr 7 '20 at 15:30
  • $\begingroup$ Perfect!! Thank you so much!! $\endgroup$
    – MathMath
    Apr 7 '20 at 15:31
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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 $F$.

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    $\begingroup$ And, as a variant on this approach to the notion of "differentiability", there are also results (from Schwartz and Grothendieck) that (on locally convex, quasi-complete TVSs) infinite-differentiability of the scalar-valued functions $x\to \lambda(f(x))$, for all $\lambda$ in the dual, implies ("strong") differentiability of the vector-valued function. $\endgroup$ Jun 26 '20 at 19:47

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