Let's say I have a row vector $\vec{x}_i \in \mathbb{R}^{1\times T}$. You can consider it as the $i$-th time-series from a set of time-series. I want to index a subsequence from $\vec{x}_i$ from $t1$ till $t2$, but I do not know how to denote it properly without causing confusion.
Can I simply denote it as $\vec{x}_i(t1:t2)$ or $\vec{x}_i |_{t1}^{t2}$?
Avoid vector arrows in print, when you are not in a geometric setting.
In elementary analysis we denote a sequence $(a_1,a_2,a_3,\ldots)$ by $\bigl(a_k\bigr)_{k\geq1}$. Similarly, when $I$ is an arbitrary index set used to parametrize a family of sets $A_\iota\,$, we denote this family by $\bigl(A_\iota\bigr)_{\iota\in I}$. Note that we are talking here about arrays, not about sets of objects. When $a_1=2$, $a_2=1$, $a_3=2$ then $$\bigl\{a_i\bigm|1\leq i\leq 3\bigr\}=\{1,2\},\qquad \bigl(a_i\bigm|1\leq i\leq 3\bigr)=(2,1,2)\ .$$ For your time series I then propose notations like $$\bigl(x_i\bigr)_{t_1\leq i\leq t_2}\ ,\qquad\bigl(x_i\bigr)_{i\in[t_1..t_2]}\ ,\qquad\bigl(x_i\bigm|t_1\leq i\leq t_2\bigr)\ .$$