I have been trying to understand this Encyclopedia of mathematics article. Specifically, in the comments section there is the following comment:
The codimension of a subspace $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all complements have the same dimension (as the orthogonal complement).
In a comment, Nate Eldredge pointed out to me that this statement is poorly written.
He provides an example of a space and a map such that the orthogonal complement of the image of the map is $0$ but the (algebraic) complement has infinite dimension.
Doesn't this mean that the comment in the Encyclopedia is not ''poorly written'' but rather plain wrong?
I apologize for hunting down the details so meticulously but I'm just a learner, not a mathematician yet and I need to know whether I really understand every detail.
For the sake of the question let's assume we have an inner product. Obviously, if we don't have one the comment is wrong. But if we do have one then the situation is less clear to me.