Let $(X,\tau)$ be a Hausdorff locally convex TVS and let $P(X)$ be a family of seminorms on $X$ that generates $\tau$. We consider the following definitions.
Definition 1. A function $f:[a,b]\to X$ is of bounded variation on $[a,b]$ if for each $p\in P(X)$, $$TV_{a}^b(f)_p=\sup\left\{\sum_{i=1}^n p(f(x_i)-f(x_{i-1}))\right\}<\infty$$ where the supremum is taken over all divisions $D=\{[x_{i-1},x_i]\}_{i=1}^n$ of $[a,b].$
Definition 2. A function $f:[a,b]\to X$ is of bounded semi-variation on $[a,b]$ if for each $p\in P(X)$, $$STV_{a}^b(f)_p=\sup\left\{p\left(\sum_{i=1}^n \alpha_i[f(x_i)-f(x_{i-1})]\right)\right\}<\infty$$ where the supremum is taken over all divisions $D=\{[x_{i-1},x_i]\}_{i=1}^n$ of $[a,b]$ and $\alpha_i\in \mathbb{R}, i=1,\cdots,n$ with $|\alpha_i|\le 1.$
I have already checked that Definition 1 implies Definition 2. I would like to see an example of a function (if possible) showing that Definition 2 does not imply Definition 1. Many thanks in advance...