# Please give me an example of closed subspace of banach space under some conditions

Please give me one example of Banach space $X$ and its closed subspaces $S,T,U$ which suffice following conditions.

Any of $S+T,T+U,U+S$ is not a closed subspace of $X$.

I can say there are some examples of Banach space $X$ and its closed subspaces $S,T$. ($S+T$ is not a closed subspace of $X$.)

one example:

$X=\{(y,z)|y \in C[0,1],z \in C[0,1]\}$, and its norm is $||(y,z)||=||y||_\infty+||z||_\infty.$

$S=\{(y,y')|y \in C^1[0,1]\},\ T=\{(0,z)|z \in C[0,1]\}.$

However, in case of three spaces, I have no idea. I would be really thankful if someone can help me.

## 1 Answer

Let $(V, ||\cdot ||)$ be a Banach space and $V_1$, $V_2$ two closed subspaces of $V$ such that $V_1 + V_2$ is not closed. Consider a Banach space $X=V\times V\times V$ with norm $||(x,y, z) ||= \max\{||x|| ,||y||, ||z||\}$ and subspaces $S=V_1 \times V_1 \times V_2 ,$ $T=V_2\times V_1 \times V_2 ,$ $U=V_2 \times V_2 \times V_1 .$ Then the subspaces $S+T ,T+U , U+S$ are not closed.