Let $K$ be a compact linear operator on the Hilbert space $H$. Prove that, for every closed subspace $V \subset H$, the $\text{Image}(I−K)(V) = \{x−Kx; x \in V \}$ is a closed subspace of $H$.
my attempt:
let $R\equiv \text{Image}(I−K)(V)$, take an arbitrary cauchy sequence $x_n-Kx_n \in R$. Since $K$ is compact operator on closed subset of Hilbert space $H$. By fredholm theorem we have that $Range(I-K)(H)$ is closed , hence complete.
So $x_n-Kx_n \to y \in Range(I-K)(H)$. Since $K$ is compact , hence it is continuous on $H$, so $(x_n-Kx_n) \to (x-Kx)$. Given that $x\in V$, then $(x-Kx)\in R$, so it must be that $(x-Kx)=y$.