Let $X$ be a normed space and $Y_1$ , $Y_2$ complete spaces and exist two injective linear application $f_1:X \to Y_1$, $f_2:X\to Y_2$
such that $\text{closing } f_1(X)=Y_1$, $\text{closing } f_2(X)=Y_2$ Show which exist an isomorphism $Y_1 $ between $Y_2$
Sorry for my english, I'm learn little to little. For true, I only want to know if it's correctly made the problem . and if is possible a guide for proof. Thank so much.
The problem is that $Y_1$ and $Y_2$ may have two different norms so $Y_1$ need not to be isomorphic to $Y_2$. Consider for example $c_{00}$-the space of all sequences $(a_n)$ such that $a_k=0$ for large $k$ and $Y_1=\ell^1$, $Y_2=\ell^2$ (the space of all summable/square summable sequences). Let $f_1,f_2$ be the natural inclusions: then all your conditions are fulfilled but of course $Y_1$ and $Y_2$ are not isomorphic.