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Let $X,Y$ be two normed linear spaces, $T_n:X\rightarrow Y$ be a sequence of isometric isomorphisms, and let $T_n\rightarrow T,$ where $T\in B(X,Y).$ This conditions implies $T$ is an isometry. Now, can we show $T$ is on-to? (which implies $T:X\rightarrow Y$ is an isometric isomorphism)

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  • $\begingroup$ In what sense does $T_n \to T$? Are you willing to assume completeness of $Y$. I don't see any hope of this without completeness. $\endgroup$ – Kavi Rama Murthy Mar 5 at 10:09

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