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Let X banach space and $T:X \to X$ compact operator with rank not finite. I want to show that $0 \in \overline{T(S_{X})}$. my idea is to suppose that $0 \notin \overline{T(S_{X})}$ then if I can show that $T(X)$ is closed ended, because i can use this result.

My problem now is to show that if $0 \notin \overline{T(S_X)}$ then $T(X)$ is closed, Can someone help me show this?

Lucas
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1 Answers1

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If $0\notin\overline{T(S_X)}$, then there exists $\varepsilon>0$ such that $T(S_X)\cap B_\varepsilon(0)=\varnothing$. Hence $\lVert T(x)\rVert\geq\varepsilon\lVert x\rVert$ for all $x\in X$, so $T$ is left-invertible. But that means $T(X)$ is closed and complemented in $X$.

user10354138
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