We know that a set is closed if and only if every convergent sequence with elements in the set has a limit point in the set.
I am reading a paper, and the paper claims that the following is due to S is a closed subspace:
Suppose we know:
S is a closed constraints subspace. K is a matrix.
- $K \in S$
- $K(GK)^n \in S, \forall n$
then
$\sum_{n=0}^\infty K(GK)^n \in S$
(Note: $\sum_{n=0}^\infty K(GK)^n$ is a convergent summation and $(GK)^n$ is convergent sequence)
Why is this related to the condition that $S$ is a closed subspace?
(This is the sum of this sequence.)
Paper title: A Characterization of Convex Problem in Decentralized Control
I suppose you are talking about some linear space. In this case, we know that the partial sums $$P_m := \sum_{n=0}^{m} K(GK)^n$$ are in $S$, because they are just finite sums of elements in $S$. If the limit $P = \lim_{m \to \infty} P_m$ exists, we know that $P \in S$ since $S$ is closed.
However, I believe that there is some kind of convergence argument missing that tells you that $P_m$ is a convergent sequence.