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?