Suppose $X$ is a topological vector space, $Y$ and $Z$ are two subspaces of $X$, and $X$ is the direct sum of $Y$ and $Z$, write $X=Y\oplus Z$. $\{x_n\}\subset X$ and $x_n=y_n+z_n$ where $y_n\in Y$ and $z_n\in Z$. Then if $x_n\rightarrow0$ as $n\rightarrow\infty$, can it concludes that $y_n\rightarrow0$ and $z_n\rightarrow0$ as $n\rightarrow\infty$?
The $x_n\rightarrow0$ as $n\rightarrow\infty$ here means $\forall U\in\mathbb{V}(0)$, i.e. $U$ is a neighborhood of $0$, $\exists N>0$, s.t. $\forall n>N$, $x_n\in U$. I guess the above conclusion is correct, but I can't proof it, and I can't find a counterexample either. It is obviously that if $X$ is a Banach space, it implies that the projection maps $P_Y$ and $P_Z$ are both continuous by Closed Graph Theorem, and the conclusion is proven out. However, in a Banach space, there defines a norm, and the convergence of a sequence can be defined by the norm. But a topological vector space can not always be normed. So what I'm thinking about here is a more general case.
Intuitively, the conclusion might be easy to be proved, but proving it is not so simple. May be it misses some conditions, such as "$X$ is locally convex" or "There is a norm or semi-norm on $X$"? I have no idea.