# Refrence for closedness of image of a compact operator

Let $$\mathcal{H}$$ be a Hilbert space, and let $$T\in K(\mathcal{H})$$ be a compact operator. There exists a theorem in the following way: "$$T(\mathcal{H})$$ is closed in $$\mathcal{H}$$ if, and only if, $$\dim(T(\mathcal{H}))<\infty$$."

Can anybody give me a reference for this theorem, please?

• See this. – David Mitra May 16 at 9:20
• @DavidMitra Thanks. I know its proof. Just need its reference. – niki May 16 at 12:34

This is an easy conseqeuence of Open Mapping Theorem. If $$T(H)$$ is closed we can think of $$T$$ as a surjective operator from $$H$$ to $$T(H)$$. By Open Mapping Theorem it is an open map. Hence the image of the closed unit ball contains some open ball around $$0$$ in $$H$$. Since $$T$$ is compact this open ball is relatively compact and this makes $$T(H)$$ finite dimensional.