I believe I've disproved this statement: If $X$ and $Y$ are subspaces of a vector space $V$ and $X \oplus Y \cong V$ (where $\oplus$ denotes the direct sum), then is it true that $V$ is the internal direct sum of $X$ and $Y$?
In disproving this statement, I found an example of subspaces $X,Y \subset V$ that share a basis element and $V \cong X \oplus Y$. By definition, to take an internal sum of two vector subspaces, one needs that the subspaces have trivial intersection. Since $X$ and $Y$ do not have trivial intersection, it does not even make sense to talk about their internal direct sum.
Which is more correct/logically sound to say:
a) $V$ is not the internal direct sum of $X$ and $Y$ or
b) $V$ is not an internal direct sum of $X$ and $Y$
I'm leaning towards the second option. I feel like using "the" guarantees existence of the object $X \oplus_{I} Y$ (where $\oplus_{I}$ denotes internal direct sum), whereas using "an" compares $V$ to an internal direct sum property-wise (i.e. is $V$ spanned by $X$ and $Y$? Is $X \cap Y = \lbrace 0 \rbrace$?).
It's the internal direct sum. There is no choice here: $V$ is the internal direct sum of $X$ and $Y$ if $V=X+Y$ and $X\cap Y=\{0\}$.