I have $\Bbb{R}^3$ domain, so it's a 3-dimensional Cartesian coordinate system with $x, y, z$ axes. I need to define a set in this domain that contains only vectors $\mathbf{v}$ that consists of components $v_x, v_y, v_z$ with limitations: $$ x_{min} < v_x < x_{max}; y_{min} < v_y < y_{max}; z_{min} < v_z < z_{max}. $$
I've started with $S = \{\mathbf{v} \in \Bbb{R}^3 | x_{min} < v_x < x_{max}, y_{min} < v_y < y_{max}, z_{min} < v_z < z_{max}\}$
I don't think it's correct. Also, it's a very long string. What if I need to define set in $\Bbb{R}^{10}$?
How to define the set?
P.S. Feel free to point out to my another mistakes in defining problem.
What you wrote is correct.
A shorter way to write it would be $$S=[x_{\rm min},x_{\rm max}]\times[y_{\rm min},y_{\rm max}]\times[z_{\rm min},z_{\rm max}]$$
For a vector in $\mathbb R^{10}$ you might go for $$ S = \prod_{n=1}^{10} [x_{n,\rm min},x_{n,\rm max}] $$
If all lower/upper limits are the same, then you can also use a notation like $$ S = [x_{\rm min},x_{\rm max}]^3 $$