Does a family of seminorm induce by a vector space isomorpism turn one of this vector space into a frechet space?

Question

Suppose $F$ is a Frechet space induced by a family of seminorms $P$. Now let $V$ be another vector space which is isomorphic to $F$ by $\tau: V \rightarrow F $. We define family of seminorms $K$ on $V$ by $$k_i=p_i\circ\tau$$

where $k_i \in K$ and $p_i \in P$

My question is , does this family of seminorms $K$ turn $V$ in a Frecht space?