Closed but not topological complementary spaces

Question

I’ve shown that if $U$ and $V$ are topologically complementary then they are closed on a normed space. Also, I’ve shown that if $X$ is a Banach space and $U,V$ are closed complementary subspaces then they are topologically complementary.

My question is what if $X$ is not Banach for second? I could not find a counter example for complementary subspaces that are closed but not topological.

My definition for topologic complements on a normed space : $U,V \subset X$ are complementary subspaces. $\forall x \in X, \quad x=u_x+v_x$ where $u_x \in U, v_x \in V$ If the mappings $P_U(x)=u_x, P_V(x)=v_x$ are continuous then $U,V$ are topologically complementary.

Thanks in advance

Answer 1

A closed subspace $U$ of a Banach space $Z$ has another closed subspace $V$ as a quasi-complement, when $U\cap V = \{0\}$ and the subspace $U+V$ is dense in $X$. Murray proved that every closed subset of a separable Banach space has a quasi-complement.

F. J. Murray, Quasicomplements and closed projections in reflexive Banach spaces, Trans. Amer. Math. Soc. 58 (1945), 77-95.

G. W. Mackey, Note on a theorem of Murray, Bull. Amer. Math. Soc. 52 (1946), no. 4, 322-325.

Now, you may take a subspace $U$ of a separable Banach space $Z$ that is not (topologically) complemented. Let $V$ be a quasi-complement of $U$ in $Z$. For your normed space, you may now take $X=U+V$ considered as a subspace of $Z$.

A concrete example would be $Z=C[0,1]$ and $U$ any infinite-dimensional reflexive subspace of $Z$.