Is the intersection of two subspaces nontrivial in an infinite dimensional vector space?

Question

Let $V = \mathbb C^{\mathbb N}$ be a vector space equipped with ordinary inner product. The notation $\mathbb C^{\mathbb N}$ means the vector space is the Cartesian product of countably infinite copies of $\mathbb C$. Let $U \subset V$ be a subspace with finite dimension $n$ and $W \subset V$ be a subspace with finite codimension $m$, i.e., $\dim( V^{\perp}) = m$. Suppose $m < n$. Is it possible that the intersection $U \cap W$ is nontrivial? Thinking in terms of finite dimensional space, it seems obvious but I am not sure whether I am missing some subtle points in infinite setting.