# Closedness of all isometric isomorphism operators in the set of all bounded linear operators on normed linear spaces

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)

• 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. – Kavi Rama Murthy Mar 5 at 10:09