Given a Hilbert space $\mathcal{H}$.
Consider a dense domain: $$\mathcal{D}\leq\mathcal{H}:\quad\overline{\mathcal{D}}=\mathcal{H}$$
Regard a closed subspace: $$\mathcal{S}\leq\mathcal{H}:\quad\mathcal{H}=\mathcal{S}\oplus\mathcal{S}^\perp$$
Then it may happen: $$\mathcal{D}\cap\left(\mathcal{S}\oplus\mathcal{S}^\perp\right)\supsetneq\mathcal{D}\cap\mathcal{S}+\mathcal{D}\cap\mathcal{S}^\perp$$ Does someone have an example at hand?
Let $\varepsilon_n,\ n\in\mathbb N$ be an ONB in $\mathcal H,$ $\varphi=\sum_n \frac 1n \varepsilon_n,$ and $\mathcal D$ be the linear span of $\{\varepsilon_n\}_{n\in \mathbb N}$ and $\varphi$. Further, let $\mathcal{S}$ be the closed linear span of $\{\varepsilon_{2k}\}_{k\in \mathbb N}.$ Then $\mathcal D\cap\mathcal{S}+\mathcal D\cap\mathcal{S}^\perp$ is a linear span of $\varepsilon_n,\ n\in\mathbb N$, which does not not contain $\varphi$.
