Suppose that $A, B, C$ are subspaces of the Hilbert space $H$. For any subspaces $X$ and $Y$, denote the orthogonal complement of $Y$ in $X$ by $X\ominus Y$. If
$$ A=B\ominus C$$
Then how to prove
$$C=B\ominus A ?$$
I do not understand is the orthogonal complement of $X$ in $Y$. Is it $X\cap Y^{\perp}$?
The subspaces have to be closed. An easy counter-example is where $A$ and $B$ are closed but $C$ is not.
Orthogonal complement of $X$ in $Y$ is $Y \cap X^{\perp}$. It is given that $A=B \cap C^{\perp}$ and this gives $B \cap A^{\perp}=B \cap [B \cap C^{\perp}]^{\perp}$. Note that $B \cap [B \cap C^{\perp}]^{\perp}\subset [C^{\perp}]^{\perp}=C $. So $B \cap A^{\perp} \subset C$ You cannot prove the reverse inclusion because there is nothing to guarantee that $C \subset B$.