Let $(X,\tau)$ be a locally convex topological vector space and denote the product space

$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$

If we endow $X^{\infty}$ with the topology induced by projection, i.e. for $\{x^n\}_{n\geq 1}\subset X^{\infty}$ and $x\in X^{\infty}$ then

$$x^n\to x\Longleftrightarrow x^n_i\to x_i,~ \forall i\geq 1$$

My question is whether $X^{\infty}$ is a locally convex space? Maybe my question is naive, but please let me know if someone knows the result. Many thanks!