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Let $X$ be an infinite dimensional Banach space. Does there always exist a compact linear operator on $X$ with infinite dimensional range? I can see that the answer is yes if $X$ is a Hilbert space, as one can take an orthonormal basis containing a countable orthonormal set $\{e_n\}_{n=1}^\infty$, and define a linear operator $T$ so that $T(e_n)=\frac1n e_n$, and zero on the other basis vectors. But I don't see how to construct one on a general Banach space. Any help would be greatly appreciated!

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