When studying functional analysis I’ve been introduced to the concept of algebraic and topologically complementary spaces. Sadly, my textbook offers no example of two subspaces which are algebraically complementary but not topologically.
Now if $X$ is, for convenience, a normed linear space with subspaces $U,V$, then $U$ and $V$ are said to be algebraically complementary if any $x \in X$ has a unique decomposition $x = u_{x}+v_{x}$ with $u_{x} \in U, v_{x} \in V$ If in addition the now well defined mappings $x \mapsto u_{x}, x\mapsto v_{x}$ are continuous, they are also said to be topologically complementary.
It is known that if two subspaces are topologically complementary, they are necessarily both closed. So I do know that for an example we should look at non closed subspaces.
As I’m only familiar with normed spaces, and not yet with general topological vector spaces, it’d help if the example is a normed space.
An easy and educational example would be greatly appreciated!
Edit: I think I’ve found a bit of a non constructive example. Let $\mathbb{H}$ be any infinite dimensionalHilbert space with an orthonormal basis $(e_{n})_{n \in \mathbb{N}} $ and consider $U= span\{ (e_{n})_{n \in \mathbb{N}} \}$ the subspace of finite linear combinations of our ONB. Invoking Zorn’s lemma one can find that it has an algebraically complementary subspace $V$, but $U$ is easily seen to be not closed and therefore it can not have a topologically complementary subspace, is this example correct?
