Let $\mathscr{H}$ be a Hilbert space and $\mathscr{M}$ be a closed linear subspace of $\mathscr{H}$.
Suppose that $\mathscr{M}+\mathscr{M^{\bot}} \equiv \{(f,g) : f \in \mathscr{M} , g \in \mathscr{M^{\bot}} \}$ and $T : \mathscr{M}+\mathscr{M^{\bot}} \to \mathscr{H}$ is defined by $T(f,g)=f+g$. I want to prove that $T$ is homeomorphism if $\mathscr{M}+\mathscr{M^{\bot}}$ is given the product topology. (This is usually phrased by stating the $\mathscr{M}$ and $ \mathscr{M^{\bot}}$ are topologically complementary in $\mathscr{H}$.)
I prove that $T$ is bijection but I can not to prove that $T$ and $T^{-1}$ are continuous. I think the definition of continuous map between topological spaces is useful i.e , for every open set $K$ of $\mathscr{H}$ the preimage $T^{-1} K$ be open set of $\mathscr{M}+\mathscr{M^{\bot}}$. I think $T^{-1}K$ is open in $\mathscr{M}+\mathscr{M^{\bot}}$ if $T^{-1}K = U \times V$ such that $U$ is an open set of $\mathscr{M}$ and $V$ is an open set of $ \mathscr{M^{\bot}}$. But I can not to comlelete this reasoning.
Define a norm on $\mathscr{M}+\mathscr{M^{\bot}}$ by
$||(f,g)||_0:=||f||+||g||$, where $|| \cdot||$ is the norm on $\mathscr{H}$.
The topology on $\mathscr{M}+\mathscr{M^{\bot}}$ is induced by $|| \cdot||_0$.
$T$ is linear and $||T(f,g)|=||f+g|| \le ||f||+||g||=||(f,g)||_0$, hence $T$ is continuous.
The open mapping theorem gives that $T^{-1}$ is continuous.