In many books and articles one finds sentences like this: "let $A$ be a closed subspace of ...".
Now my question might be stupid, but I am always wondering what they mean by closed subspace? Is this meant topologically, i.e. is $A$ a subspace such that every convergent sequence has a limit in $A$ or is this meant as a vector space property, i.e. if $A$ is a subset of the vector space $(B,\oplus,\odot)$, then $A$ is closed w.r.t. $\oplus$ and $\odot$.
Is there a way to tell which of these is meant by the author of a book or an article?
In case my question is not quite clear, let me give an example:
A multiresolution analysis consists of a sequence of closed subspaces $V_j\subset L^2(\mathbb R)$ satisfying ..."
Now as $L^2$ is a vector space, I am not sure if this is meant topologically or as a vector subspace property.
Any input on this is welcome. Thanks!
I have never seen "closed subspace" used to mean "subspace closed wrt the vector operations". This makes sense because the property of being closed wrt the vector operations is part of the meaning of the word "subspace", so it would be redundant to specify it by an adjective, while the property of being topologically closed is not redundant, and is, moreover, a frequently important property in Banach space theory. (For example, this is the property that makes the quotient space a Banach space.)