Can we show that if 2 spaces defined with a norm are isomorphic and one of them is Banach, then the other space is also Banach?

Question

I am asking this because I have a space V= {$p(x)$| $p(x)$=$ax$$^3$, a $\in \mathbb{R}$ } with norm ${\lVert}p(x){\rVert}= \lvert a \rvert $ (i.e the space of all monomials of degree 3) and I've been asked to show that it's Banach. Of course I could use the approach that it follows that every finite dimensional normed space is Banach. I did however want to know that if I could show that if V is isomorphic to $\mathbb{R}$ then V is Banach as $\mathbb{R}$ is Banach.