Does there exist any surjective continuous linear map $T:l^2 \to l^1$ ? I believe such a map does not exist ; my reasoning is like the existence of such $T$ would imply $T^* : l^1{^*}\cong l^{\infty} \to l^2{^*}\cong l^2$ continuous and injective , but then I am stuck . Is my approach correct or is there any other way ? Please help . Thanks in advance
Does there exist any surjective continuous linear map $T:l^2 \to l^1$?
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functional-analysis
banach-spaces
normed-spaces
2 Answers
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The open mapping theorem implies that $T$ is a quotient map. As quotients of Hilbert spaces are Hilbert, $\ell^1$ would have an equivalent Hilbert norm and, in particular, it would be reflexive which is not the case.
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If there were one, the adjoint of such map would embed $\ell_\infty$ into $\ell_2$, which is nonsense as the former space is non-separable.