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I'm reading through a text that has been making references to "(not necessarily orthogonal) direct sums" of Hilbert spaces. What would a non-orthogonal direct sum be? Is that something like a direct sum $A \oplus B$ where $A = C \oplus D$ and $B = C \oplus E$? That is, $ (C \oplus D) \oplus (C \oplus E). $

What are some caveats to watch out for when we have a non-orthogonal direct sum?

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A non-orthogonal direct sum of Hilbert spaces is a Hilbert space $A \oplus B$ such that $A$ is not orthogonal to $B$; that is, there are vectors $a$ in $A$ and $b$ in $B$ such that $\langle a , b \rangle \neq 0$.

For example, the Hilbert space $\mathbb{R}^2$ (with the standard inner product) is the orthogonal direct sum of $\{ (x,0) : x \in \mathbb{R} \}$ and $\{ (0,y) : y \in \mathbb{R} \}$. It is also the non-orthogonal direct sum of $\{ (x,0) : x \in \mathbb{R} \}$ and $\{ (y,y) : y \in \mathbb{R} \}$.