The problem statement, all variables and given/known data.
Let ##E## be a Banach space and let ##S,T \subset E## two closed subspaces. Prove that if dim## T< \infty##, then ##S+T## is also closed.
The attempt at a solution.
To prove that ##S+T## is closed I have to show that if ##x## is a limit point of ##S+T##, then ##x \in S+T##. Let ##x## be a limit point of ##S+T##. Then, there is a sequence ##\{x_n\}_{n \in \mathbb N} \subset S+T## : ##x_n→x##. For each ##n \in \mathbb N##, ##x_n## is in ##S+T##, this means that ##x_n=s_n+t_n## for some ##s_n \in S## and some ##t_n \in T##. ##\{s_n\}_{n \in \mathbb N} \subset S## and ##\{t_n\}_{n \in \mathbb n} \subset T##. Here I got stuck, I would like to show that these two sequences are convergent. I have to use all the facts I know (the completeness of ##X##, that ##S## and ##T## are closed and that ##T## is finite dimensional), but I don't know how.
To be honest, I don't even understand intuitively why this statement is true. I mean, why one just requires for ##T## to be finite dimensional and not both of them? Why infinity can screw up closure with the sum if the two subspaces are closed?
Let ##E## be a Banach space and let ##S,T \subset E## two closed subspaces. Prove that if dim## T< \infty##, then ##S+T## is also closed.
The attempt at a solution.
To prove that ##S+T## is closed I have to show that if ##x## is a limit point of ##S+T##, then ##x \in S+T##. Let ##x## be a limit point of ##S+T##. Then, there is a sequence ##\{x_n\}_{n \in \mathbb N} \subset S+T## : ##x_n→x##. For each ##n \in \mathbb N##, ##x_n## is in ##S+T##, this means that ##x_n=s_n+t_n## for some ##s_n \in S## and some ##t_n \in T##. ##\{s_n\}_{n \in \mathbb N} \subset S## and ##\{t_n\}_{n \in \mathbb n} \subset T##. Here I got stuck, I would like to show that these two sequences are convergent. I have to use all the facts I know (the completeness of ##X##, that ##S## and ##T## are closed and that ##T## is finite dimensional), but I don't know how.
To be honest, I don't even understand intuitively why this statement is true. I mean, why one just requires for ##T## to be finite dimensional and not both of them? Why infinity can screw up closure with the sum if the two subspaces are closed?
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