Orthogonal complement of subspaces of Hilbert space

Question

Let $H$ be a complex, infinite dimensional, separable Hilbert space

Let $\{h1_m\}_{m \in \mathbb{N}}$ be a sequence in $H$

Let $\{h2_m\}_{m \in \mathbb{N}}$ be a sequence in $H$

Let $V1=\operatorname{Span}(\{h1_m\}_{m \in \mathbb{N}})$

Let $V2=\operatorname{Span}(\{h2_m\}_{m \in \mathbb{N}})$

We know that $\{h1_m\}_{m \in \mathbb{N}} \cap \{h2_m\}_{m \in \mathbb{N}}$ is linearly independent algebraically (for finite linear combinations)

Let $U1=\overline{V1}$

Let $U2=\overline{V2}$

Let $W= U1 \cap U2$

Let $W1$ be the orthogonal complement of subspace $W$ of the space $U1$

Let $W2$ be the orthogonal complement of subspace $W$ of the space $U2$

My questions:

Is it true that

$V1 \subset W1$ and $V2 \subset W2$

Thanks.